Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | #39725 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1660 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)(3,7)(4,6)(5,8)(17,18,23,27,21,19,22,25)(20,24,30,32,28,26,29,31);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(18,19)(24,26)(25,27)(31,32);(3,8)(5,7)(10,13)(14,15)(17,19,21,18)(20,26,28,24)(22,25,23,27)(29,31,30,32);(2,6)(5,7)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32);(9,11,12,16)(10,14,13,15)(20,28)(24,26)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=gi,b^2=fh,c^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6710024);