Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151057 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2158 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #12834 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×M16 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
232 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 96 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)(3,15)(4,22,24,10)(5,27)(6,31)(7,34)(8,12)(9,18)(11,16)(13,28)(14,32)(17,55,57,45)(19,63)(20,66)(21,47)(23,71,73,78)(25,77,75,84)(26,41,76,70)(29,83)(30,79)(33,85)(35,56,67,108)(36,62)(37,90)(38,91)(39,51,68,101)(40,72,69,119)(42,54)(43,93)(44,82)(46,50)(48,60)(49,64)(52,81)(53,80)(58,112,110,114)(59,96,111,106)(61,86)(65,115)(74,121,102,120)(87,122,116,128)(88,123,117,109)(89,124)(92,125,118,103)(94,99,104,126)(95,107,105,127)(97,100)(98,113);(1,3,16,50,6,19,52,91,32,64,93,124,82,113,30,62,8,21,54,100,33,65,29,61,7,20,53,90,28,60,5,18)(2,9,27,48,13,37,80,66,34,86,83,115,85,97,42,47,12,36,79,98,44,89,43,49,14,38,81,63,31,46,11,15)(4,17,51,99,25,58,102,123,76,111,125,128,119,127,73,108,24,57,101,126,75,110,74,109,26,59,103,122,72,107,23,56)(10,35,78,95,40,87,118,106,70,117,120,112,77,104,68,55,22,67,71,105,69,116,92,96,41,88,121,114,84,94,39,45);(1,4)(2,10)(3,17)(5,23)(6,25)(7,26)(8,24)(9,35)(11,39)(12,22)(13,40)(14,41)(15,45)(16,51)(18,56)(19,58)(20,59)(21,57)(27,78)(28,72)(29,74)(30,73)(31,84)(32,76)(33,75)(34,70)(36,67)(37,87)(38,88)(42,68)(43,92)(44,69)(46,94)(47,55)(48,95)(49,96)(50,99)(52,102)(53,103)(54,101)(60,107)(61,109)(62,108)(63,114)(64,111)(65,110)(66,106)(71,79)(77,85)(80,118)(81,121)(82,119)(83,120)(86,117)(89,116)(90,122)(91,123)(93,125)(97,104)(98,105)(100,126)(112,115)(113,127)(124,128)Polycyclic presentation:
2^9|a^2=f,b^2=dg,d^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72320480, #72320831, #72320868, … |
| maximal normal subgroup of | #72320480, #72320831, #72320868, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59490);