Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7207866 |
| 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 | #53064 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1466 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)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,23,30,26)(20,31,33,35)(21,25,34,40)(24,37,39,28)(29,36,43,41)(32,44,45,46)(38,47,48,42);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,30)(18,23,27,26)(20,28,33,37)(21,29,34,43)(24,35,39,31)(25,36,40,41)(32,42,45,47)(38,46,48,44);(2,6)(5,7)(11,15)(13,16)(17,20,30,37,22,33,19,28)(18,24,23,35,27,39,26,31)(21,32,43,47,34,45,29,42)(25,38,36,46,40,48,41,44);(3,8)(5,7)(10,14)(11,15)(17,21)(18,25)(19,29)(20,32)(22,34)(23,36)(24,38)(26,41)(27,40)(28,42)(30,43)(31,44)(33,45)(35,46)(37,47)(39,48);(2,6)(5,7)(11,15)(13,16)(21,34)(25,40)(29,43)(32,45)(36,41)(38,48)(42,47)(44,46)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7115581);