Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,36)(30,38)(31,39)(32,40)(33,42)(34,41)(35,37)(43,46)(44,48)(45,47);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19,23,31)(18,24,28,39)(20,29,33,43)(21,30,22,35)(25,36,42,46)(26,38,27,37)(32,44,34,45)(40,48,41,47);(10,14)(11,16)(17,20)(18,25)(19,29)(21,34)(22,32)(23,33)(24,36)(26,41)(27,40)(28,42)(30,45)(31,43)(35,44)(37,48)(38,47)(39,46);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21)(18,26)(20,32)(22,23)(25,40)(27,28)(29,43)(30,35)(33,34)(36,46)(37,38)(41,42);(2,7)(6,8)(10,14)(13,15)(17,21,23,22)(18,27,28,26)(19,30,31,35)(20,32,33,34)(24,37,39,38)(25,41,42,40)(29,44,43,45)(36,47,46,48)Polycyclic presentation:
2^9|b^2=fi,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7456452);