Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,3,4)(5,6,8,7)(10,14)(15,16)(17,18)(19,26)(20,28)(21,25)(22,23)(24,30)(27,31)(29,32);(6,7)(9,10,13,14)(11,15,12,16)(17,19,21,22)(18,23,25,26)(20,29,30,27)(24,32,28,31);(2,4)(5,6,8,7)(9,11,13,12)(10,15,14,16)(17,20,21,30)(18,24,25,28)(19,27,22,29)(23,31,26,32);(5,7,8,6)(9,12,13,11)(10,15,14,16)(18,23)(19,22)(24,31)(25,26)(27,29)(28,32)Polycyclic presentation:
2^9|a^2=fg,b^2=egi,c^2=e,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348238);