Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle | C24 | C23×C4 |
| prime radical | C2×#2397 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2250 conjugacy classes of subgroups; only the 402 normal subgroups are shown.
This group has 2250 conjugacy classes of subgroups — too many to draw. Only its 402 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:
(2,3)(6,8)(13,14,17,21,18,16,20,25)(15,19,24,27,23,22,26,28);(1,2,4,3)(9,10,11,12)(13,15)(14,19)(16,22)(17,24)(18,23)(20,26)(21,27)(25,28);(5,6,7,8)(10,12)(13,16,17,25,18,14,20,21)(15,22,24,28,23,19,26,27);(1,3,4,2)(15,23)(19,22)(24,26)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=fh,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72239336, #72239410, #72265656, … |
| maximal normal subgroup of | #72239336, #72239410, #72265656, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282472);