Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| 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 | |
1598 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)(17,18,21,24)(19,26,28,23)(20,30,25,22)(27,32,31,29);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(17,19)(18,23)(20,31)(21,28)(22,29)(24,26)(25,27)(30,32);(3,8)(5,7)(10,13)(14,15)(17,20,22,18,21,25,30,24)(19,27,29,26,28,31,32,23);(2,6)(5,7)(18,24)(19,28)(20,25)(29,32);(9,11,12,16)(10,14,13,15)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=g,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6712607);