Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395154 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1518 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)(9,10)(11,16)(12,14)(13,15)(17,18,23,29)(19,30,32,26)(20,25,31,24)(21,28,22,27);(2,4)(5,6)(7,8)(17,19,22,20,23,32,21,31)(18,24,28,26,29,25,27,30);(6,8)(9,11,13,12)(10,14,15,16)(17,19,21,31,23,32,22,20)(18,25,27,26,29,24,28,30);(9,12,13,11)(10,14,15,16)(17,20,22,32,23,31,21,19)(18,26,28,25,29,30,27,24)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=fi,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264289, #72264302, #72264335, … |
| maximal normal subgroup of | #72264289, #72264302, #72264335, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 315792);