Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,8,7)(10,15)(11,19)(12,13)(16,18)(17,24)(20,21);(1,2,3,4)(5,7,8,6)(9,10,13,16,14,18,12,15)(11,17,20,23,21,24,19,22);(2,4)(10,16,18,15)(17,23,24,22);(9,11,12,19,14,21,13,20)(10,17,15,22,18,24,16,23)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=f,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72226903, #72226912, #72226917 |
| maximal normal subgroup of | #72226903, #72226912, #72226917 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274736);