Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | #34963 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,22)(16,19)(17,20)(18,21)(23,28)(24,27)(25,26);(2,4)(15,23)(16,25)(18,24)(19,26)(21,27)(22,28);(6,9)(10,12)(13,15,18,23,17,16,24,25)(14,19,21,26,20,22,27,28);(13,16,18,25,17,15,24,23)(14,19,21,26,20,22,27,28);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|c^2=h,d^2=h,e^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72283931, #72284134, #72284557, … |
| maximal normal subgroup of | #72283931, #72284134, #72284557, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374261);