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×#2619 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2090 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)(11,15)(12,13)(14,16)(18,22)(20,27)(23,28)(24,29)(26,31)(30,32);(5,6)(7,8)(10,13)(14,15)(17,18,20,22,21,24,27,29)(19,23,26,28,25,30,31,32);(2,4)(6,8)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32);(9,11,12,16)(10,14,13,15)(19,25)(23,30)(26,31)(28,32)Polycyclic presentation:
2^9|b^2=eh,d^2=h,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 | #72264689, #72264696, #72264751, … |
| maximal normal subgroup of | #72264689, #72264696, #72264751, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323081);