Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150990 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11964 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| prime radical | C2×#508 | C4 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
252 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 64 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:
(1,2,6,14,8,44,35,13,7,42,39,65,43,57,37,103,10,17,41,62,46,16,36,59,9,15,40,58,34,104,5,33)(3,18,21,63,23,85,76,64,22,83,80,51,84,110,78,50,25,67,82,49,86,56,77,105,24,60,81,11,45,48,20,75)(4,26,28,61,30,55,94,52,29,54,97,115,99,128,27,91,32,12,38,53,101,90,93,88,31,89,98,113,92,112,95,102)(19,68,70,116,72,111,122,117,71,118,123,87,125,109,69,108,74,66,79,107,126,120,121,119,73,114,124,47,100,106,96,127);(1,3,10,25)(2,11,17,51)(4,19,32,74)(5,21,37,82)(6,20,41,78)(7,24,9,22)(8,45,46,84)(12,47,26,87)(13,56,59,85)(14,60,62,83)(15,63,42,49)(16,64,44,105)(18,65,67,58)(23,43,86,34)(27,79,95,70)(28,96,38,69)(29,73,31,71)(30,100,101,125)(33,48,103,110)(35,81,36,80)(39,77,40,76)(50,104,75,57)(52,111,88,120)(53,114,61,118)(54,116,89,107)(55,117,90,119)(66,115,68,113)(72,99,126,92)(91,106,102,109)(93,123,94,124)(97,121,98,122)(108,128,127,112);(1,4)(2,12)(3,19)(5,27)(6,38)(7,29)(8,30)(9,31)(10,32)(11,47)(13,52)(14,61)(15,54)(16,55)(17,26)(18,66)(20,69)(21,79)(22,71)(23,72)(24,73)(25,74)(28,41)(33,102)(34,92)(35,93)(36,94)(37,95)(39,98)(40,97)(42,89)(43,99)(44,90)(45,100)(46,101)(48,109)(49,107)(50,108)(51,87)(53,62)(56,111)(57,112)(58,113)(59,88)(60,118)(63,116)(64,117)(65,115)(67,68)(70,82)(75,127)(76,121)(77,122)(78,96)(80,124)(81,123)(83,114)(84,125)(85,120)(86,126)(91,103)(104,128)(105,119)(106,110)Polycyclic presentation:
2^9|a^2=e,b^2=i,d^2=gh,e^2=fgh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=cfh,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gh,h^b=hi,e^c=ei44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58187);