Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394180 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#1999 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1756 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)(7,8)(18,22)(20,23)(24,28)(25,29)(27,30)(31,32);(9,10)(11,14)(12,13)(15,16)(17,18,23,29,21,25,20,22)(19,24,30,32,26,31,27,28);(2,4)(10,13)(14,16)(17,19)(18,24)(20,27)(21,26)(22,28)(23,30)(25,31)(29,32);(6,8)(9,11,12,15)(10,14,13,16)(19,26)(24,31)(27,30)(28,32)Polycyclic presentation:
2^9|b^2=ei,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264145, #72264208, #72264332, … |
| maximal normal subgroup of | #72264145, #72264208, #72264332, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311283);