Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13700 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2215 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13059 |
| prime radical | #33617 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
301 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 301 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2)(3,16,21,50)(4,23,26,49)(5,28)(6,13)(7,37)(8,40)(9,15)(10,19,43,29)(11,25,47,36)(12,17)(14,38)(18,66)(20,58,35,60)(22,78,77,114)(24,82,30,94)(27,48,87,109)(31,55)(32,90)(33,96)(34,92)(39,52)(41,99)(42,74,45,93)(44,105,104,117)(46,106,51,110)(53,63)(54,65)(56,70)(57,61,97,75)(59,73,88,76)(62,108,85,80)(64,91)(67,101)(68,100)(69,122)(71,115)(72,79)(81,83,98,86)(84,89,95,107)(102,125)(103,121)(111,127)(112,124)(113,128)(116,118)(119,123)(120,126);(1,3,17,61,39,77,33,93,6,35,63,105,38,76,5,29,9,21,65,75,7,22,32,74,8,20,64,117,31,73,34,19)(2,10,28,88,55,104,53,58,40,42,96,78,37,97,12,50,15,43,92,59,14,44,91,60,13,45,90,114,52,57,54,16)(4,18,62,116,87,124,95,99,36,68,106,128,86,123,30,67,26,71,85,125,27,69,84,121,25,70,110,127,83,120,24,72)(11,41,89,122,109,118,80,115,23,79,94,119,98,111,51,100,47,103,107,112,48,102,108,66,49,101,82,126,81,113,46,56);(1,4)(2,11)(3,18)(5,30)(6,36)(7,27)(8,25)(9,26)(10,41)(12,51)(13,49)(14,48)(15,47)(16,56)(17,62)(19,72)(20,70)(21,71)(22,69)(23,40)(24,34)(28,89)(29,67)(31,83)(32,84)(33,95)(35,68)(37,98)(38,86)(39,87)(42,79)(43,103)(44,102)(45,101)(46,54)(50,100)(52,81)(53,80)(55,109)(57,113)(58,115)(59,112)(60,66)(61,116)(63,106)(64,110)(65,85)(73,120)(74,121)(75,125)(76,123)(77,124)(78,119)(82,90)(88,122)(91,108)(92,107)(93,99)(94,96)(97,111)(104,118)(105,128)(114,126)(117,127)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=ghi,e^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=di,e^b=ei,f^b=fi,d^c=di,e^c=ei,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72320566, #72320813, #72320903 |
| maximal normal subgroup of | #72320566, #72320813, #72320903 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59837);