Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#3054 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,13,17,14,18,12,15)(11,19,22,24,20,23,21,16);(5,6)(7,8)(10,15,18,17)(11,20)(16,19,24,23)(21,22);(2,4)(6,8)(11,20)(16,24)(19,23)(21,22);(9,11,12,21,14,20,13,22)(10,16,17,19,18,24,15,23)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216198 |
| maximal normal subgroup of | #72216198 |
| maximal quotient of | #72215994 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272212);