Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| 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 ◁ C4×#125 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1565 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,6,4,8)(9,10)(11,14)(12,16)(13,15)(17,18,21,24)(19,26,29,23)(20,30,22,25)(27,31,28,32);(1,3,5,4)(2,6,7,8)(9,11,14,10)(12,16,13,15)(17,19)(18,23)(20,28)(21,29)(22,27)(24,26)(25,32)(30,31);(2,7)(6,8)(9,12,14,13)(10,15,11,16)(18,24)(19,27)(20,22)(23,31)(26,32)(28,29);(9,12,14,13)(10,15,11,16)(17,20,21,22)(18,25,24,30)(19,28,29,27)(23,32,26,31);(1,4,5,3)(2,8,7,6)(9,13,14,12)(10,15,11,16)(19,27)(20,22)(23,32)(25,30)(26,31)(28,29)Polycyclic presentation:
2^9|a^2=fgi,b^2=gi,c^2=fi,d^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6627421);