Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499712 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#2206 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
940 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 940 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,9,12)(7,11,8,10)(14,19)(15,21)(16,25)(17,24)(20,23)(22,28)(26,27);(5,7,9,8)(6,10,12,11)(14,20)(15,21)(16,25)(17,24)(19,23)(22,27)(26,28);(1,2,3,4)(5,7,9,8)(6,11,12,10)(13,14,21,27,17,22,16,20,18,23,25,28,24,26,15,19);(5,8,9,7)(6,11,12,10)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=efh,d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72286261 |
| maximal normal subgroup of | #72286261 |
| maximal quotient of | #72287727 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410729);