Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8479 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1779 |
| prime radical | #3455 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
549 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 549 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 80 points:
(1,2)(3,10,12,7)(4,6)(5,9)(8,13)(11,15,16,14)(17,18,21,27,22,28,38,50,23,26,39,47,40,48,63,71)(19,24,31,42,32,43,57,67,30,44,54,68,55,69,75,80)(20,33,35,49,36,59,61,72,34,25,37,45,60,46,62,70)(29,41,52,64,53,65,74,78,51,58,56,66,73,77,76,79);(2,6)(4,13)(7,14)(8,9)(10,15)(11,16)(17,19)(18,24)(20,29)(21,31)(22,32)(23,30)(25,41)(26,44)(27,42)(28,43)(33,58)(34,51)(35,52)(36,53)(37,56)(38,57)(39,54)(40,55)(45,64)(46,65)(47,68)(48,69)(49,66)(50,67)(59,77)(60,73)(61,74)(62,76)(63,75)(70,78)(71,80)(72,79);(1,3,5,12)(2,7,9,10)(4,11,13,16)(6,14,8,15)(17,20)(18,25)(19,29)(21,37)(22,36)(23,34)(24,41)(26,33)(27,49)(28,46)(30,51)(31,56)(32,53)(35,39)(38,62)(40,60)(42,66)(43,65)(44,58)(45,47)(48,59)(50,72)(52,54)(55,73)(57,76)(61,63)(64,68)(67,79)(69,77)(70,71)(74,75)(78,80)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dg,e^b=ei,d^c=di,f^c=fgi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36907);