Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324564 |
| 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 | #45196 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1437 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(17,18)(19,28)(20,31)(21,26)(22,27)(23,24)(25,32)(29,41)(30,44)(33,47)(34,38)(35,48)(36,43)(37,39)(40,46)(42,45);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,23)(18,24,27,28)(20,32,34,29)(21,30,36,37)(25,31,41,38)(26,39,43,44)(33,46,35,45)(40,47,42,48);(9,11)(10,13)(12,16)(14,15)(17,20,19,29,22,34,23,32)(18,25,24,38,27,41,28,31)(21,35,30,46,36,33,37,45)(26,42,39,47,43,40,44,48);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,21,22,36)(18,26,27,43)(19,30,23,37)(20,33,34,35)(24,39,28,44)(25,40,41,42)(29,45,32,46)(31,47,38,48);(2,5)(6,8)(10,14)(13,15)(18,27)(21,36)(24,28)(25,41)(30,37)(31,38)(33,35)(45,46)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=gi,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7235404);