Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395154 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(9,10,13,15)(11,16,12,14)(17,18,21,23)(19,24,22,30)(20,27,29,32)(25,26,31,28);(5,6)(7,8)(9,11,13,12)(10,14,15,16)(18,23)(24,30)(25,31)(27,32);(2,4)(6,8)(9,12,13,11)(10,14,15,16)(17,19,21,22)(18,24,23,30)(20,28,29,26)(25,32,31,27);(17,20,22,28,21,29,19,26)(18,25,24,27,23,31,30,32)Polycyclic presentation:
2^9|a^2=e,b^2=ei,c^2=e,d^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264868, #72264873, #72264897, … |
| maximal normal subgroup of | #72264868, #72264873, #72264897, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319406);