Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1646 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)(4,8)(5,6)(9,10,12,13)(11,15,16,14)(17,18)(19,24)(20,22)(21,25)(23,26)(27,28)(29,31)(30,32);(2,6)(7,8)(10,13)(11,16)(17,19,23,30)(18,24,26,32)(20,27,25,31)(21,29,22,28);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,18,22,25,23,26,21,20)(19,27,29,32,30,31,28,24);(1,4,5,3)(2,8,6,7)(10,13)(14,15)(17,20,22,18,23,25,21,26)(19,24,29,27,30,32,28,31);(2,6)(7,8)(10,13)(14,15)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=fh,d^2=fgi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,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, 7510591);