Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,4)(9,10)(11,15)(12,16)(13,14)(17,18,21,25)(19,27,29,23)(20,24,33,38)(22,26,32,37)(28,39,41,34)(30,40,43,35)(31,36,44,46)(42,47,48,45);(1,2,5,6)(3,7,4,8)(19,28)(20,31)(22,32)(23,34)(24,36)(26,37)(27,39)(29,41)(30,43)(33,44)(35,40)(38,46);(1,3,5,4)(2,7,6,8)(10,14)(15,16)(17,19,22,28,21,29,32,41)(18,23,26,34,25,27,37,39)(20,30,31,42,33,43,44,48)(24,35,36,45,38,40,46,47);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(19,29)(23,27)(28,41)(30,43)(34,39)(35,40)(42,48)(45,47);(2,6)(7,8)(9,12,13,11)(10,16,14,15)(17,20,32,44,21,33,22,31)(18,24,37,46,25,38,26,36)(19,30,41,48,29,43,28,42)(23,35,39,47,27,40,34,45)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fh,d^2=gi,e^2=fgh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6751468);