Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×#102 |
| prime radical | C23×#135 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3274 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)(5,6,9,15,11,13,10,25)(7,16,18,33,19,32,26,31)(8,20,22,30,24,35,23,14)(12,21,28,34,27,36,29,17);(5,7)(6,12)(8,21)(9,18)(10,26)(11,19)(13,27)(14,31)(15,28)(16,20)(17,23)(22,34)(24,36)(25,29)(30,33)(32,35);(2,4);(6,13)(12,27)(14,30)(15,25)(16,32)(20,35)(28,29)(31,33)Polycyclic presentation:
2^9|a^2=g,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=eh,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239819, #72239835, #72239842, … |
| maximal normal subgroup of | #72239819, #72239835, #72239842, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400715);