Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300980 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1982 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)(3,6)(4,5)(7,8)(17,18,22,26)(19,23,28,36)(20,25,31,40)(21,24,33,27)(29,35,41,47)(30,37,42,34)(32,38,44,39)(43,46,48,45);(2,5)(3,7)(10,14)(11,12)(17,19)(18,23)(20,29)(21,30)(22,28)(24,37)(25,35)(26,36)(27,34)(31,41)(32,43)(33,42)(38,46)(39,45)(40,47)(44,48);(9,10)(11,16)(12,15)(13,14)(18,24)(19,28)(20,31)(21,33)(23,34)(25,39)(26,27)(35,46)(36,37)(38,40)(43,48)(45,47);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20,22,31)(18,25,26,40)(19,29,28,41)(21,32,33,44)(23,35,36,47)(24,38,27,39)(30,43,42,48)(34,45,37,46);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(18,26)(19,28)(20,31)(24,27)(30,42)(32,44)(35,47)(45,46)Polycyclic presentation:
2^9|a^2=i,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=efi,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7395053);