Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×C22⋊C8 |
| prime radical | C23×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,15,10,16,24,33,11,13,25,29,8,14,22,31)(7,17,19,34,18,27,35,32,20,30,23,36,21,12,26,28);(5,7)(6,12)(8,21)(9,23)(10,18)(11,20)(13,27)(14,30)(15,32)(16,17)(19,25)(22,35)(24,26)(28,29)(31,34)(33,36);(2,4);(6,13)(12,27)(14,16)(15,29)(17,30)(28,32)(31,33)(34,36)Polycyclic presentation:
2^9|a^2=g,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^b=ei,g^b=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72239998, #72240009, #72240693, … |
| maximal normal subgroup of | #72239998, #72240009, #72240693, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400536);