Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2535 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6)(7,8)(13,14,18,20,16,21,17,22)(15,23,24,27,25,19,26,28);(1,2,4,3)(9,10)(11,12)(13,15,17,26,16,25,18,24)(14,19,22,27,21,23,20,28);(6,8)(10,12)(14,20)(15,24)(17,18)(19,23)(21,22)(25,26);(1,3,4,2)(14,21)(19,23)(20,22)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fi,g^c=gi,h^c=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72264274, #72264280, #72264291, … |
| maximal normal subgroup of | #72264274, #72264280, #72264291, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272103);