Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15310 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3031 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(9,10,13,12)(11,15,16,14)(17,18,21,20)(19,23,24,22)(26,28)(30,31);(10,12)(14,15)(18,20)(22,23)(25,26,29,28)(27,31,32,30);(17,19)(18,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,14,12,15)(17,19)(18,22)(20,23)(21,24)(25,28,29,26)(27,31,32,30);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,16,14)(17,20,21,18)(19,23,24,22)(25,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=fh,b^2=h,d^2=fg,e^2=hi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72238495, #72238497, #72262591, … |
| maximal normal subgroup of | #72238495, #72238497, #72262591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 593040);