Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395154 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1548 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)(11,14)(12,13)(15,16)(17,18,23,28)(19,25,30,31)(20,24,29,32)(21,27,22,26);(5,6)(7,8)(10,13)(14,16)(17,19,21,20,23,30,22,29)(18,24,26,25,28,32,27,31);(2,4)(6,8);(9,11,12,15)(10,14,13,16)(17,20,21,30,23,29,22,19)(18,25,26,32,28,31,27,24)Polycyclic presentation:
2^9|a^2=i,b^2=hi,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264155, #72264237, #72264281, … |
| maximal normal subgroup of | #72264155, #72264237, #72264281, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312761);