Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395151 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
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 | |
1392 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)(9,10,12,14)(11,15,16,13)(17,18,20,22)(19,25,26,32)(21,29,28,24)(23,31,30,27);(5,6)(7,8)(9,11,12,16)(10,13,14,15)(18,22)(23,30)(24,29)(25,32);(2,4)(6,8);(17,19,21,27,20,26,28,31)(18,23,24,25,22,30,29,32)Polycyclic presentation:
2^9|a^2=e,b^2=ei,d^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264743, #72264745, #72264835, … |
| maximal normal subgroup of | #72264743, #72264745, #72264835, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319108);