Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309060 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1611 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)(3,7)(4,6)(5,8)(17,18,23,29,22,28,21,27)(19,30,33,45,34,24,38,42)(20,35,36,44,37,26,39,25)(31,46,48,40,32,41,47,43);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,34)(18,24,28,30)(20,32,37,31)(21,38,23,33)(25,43,44,40)(26,41,35,46)(27,45,29,42)(36,47,39,48);(9,11)(10,13)(12,16)(14,15)(17,20,22,37)(18,25,28,44)(19,31,34,32)(21,36,23,39)(24,40,30,43)(26,27,35,29)(33,47,38,48)(41,42,46,45);(3,8)(5,7)(10,14)(11,16)(17,20)(18,26)(19,32)(21,39)(22,37)(23,36)(24,41)(25,29)(27,44)(28,35)(30,46)(31,34)(33,47)(38,48)(40,45)(42,43);(2,6)(5,7)(10,14)(13,15)(17,21,22,23)(18,27,28,29)(19,33,34,38)(20,36,37,39)(24,42,30,45)(25,35,44,26)(31,47,32,48)(40,46,43,41)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7216329);