Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8510 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1779 |
| prime radical | #3529 | C4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
575 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 72 points:
(2,5)(3,7)(6,8)(9,10,14,21,16,22,38,51,15,19,37,45,13,20,34,48)(11,23,26,49,25,46,58,70,27,17,35,41,32,40,62,67)(12,28,31,60,30,44,61,50,29,47,36,63,33,18,42,43)(24,53,56,71,57,52,69,68,54,65,59,72,55,39,64,66);(1,2)(3,8)(4,6)(5,7)(9,11)(10,17)(12,24)(13,32)(14,35)(15,27)(16,25)(18,39)(19,23)(20,46)(21,49)(22,40)(26,37)(28,53)(29,54)(30,57)(31,59)(33,55)(34,58)(36,56)(38,62)(41,45)(42,69)(43,68)(44,52)(47,65)(48,67)(50,66)(51,70)(60,72)(61,64)(63,71);(3,7)(5,8)(9,12)(10,18)(11,24)(13,33)(14,36)(15,29)(16,30)(17,39)(19,44)(20,47)(21,50)(22,28)(23,52)(25,57)(26,59)(27,54)(31,37)(32,55)(34,61)(35,56)(38,42)(40,53)(41,68)(43,45)(46,65)(48,60)(49,66)(51,63)(58,64)(62,69)(67,72)(70,71)Polycyclic presentation:
2^9|a^2=f,d^2=gi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dgi,e^b=eg,f^b=fi,d^c=dg,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36860);