Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C2×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C8×D4 |
| prime radical | #3050 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
698 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)(5,6,10,16,11,15,21,31,12,17,20,13,9,26,7,18)(8,14,23,29,24,28,34,36,25,30,33,27,22,35,19,32);(5,7,9,20,12,21,11,10)(6,13,15,18,17,16,26,31)(8,19,22,33,25,34,24,23)(14,27,28,32,30,29,35,36);(2,4);(5,8,10,23,11,24,21,34,12,25,20,33,9,22,7,19)(6,14,16,29,15,28,31,36,17,30,13,27,26,35,18,32)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=g,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei176 conjugacy classes — view the full character table.
| maximal subgroup of | #72286922, #72287227, #72287229, … |
| maximal normal subgroup of | #72286922, #72287227, #72287229, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402992);