Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1657 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6,8,7)(9,10)(11,13)(12,14)(15,19)(16,20)(17,21)(18,24)(22,23);(1,2,4,3)(9,11,14,21,16,22,15,18)(10,17,19,13,20,24,12,23);(9,12,15,20,16,19,14,10)(11,17,18,23,22,24,21,13);(1,3,4,2)(5,7,8,6)(9,13,14,24,16,23,15,17)(10,18,19,11,20,21,12,22)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fi,d^2=eh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=di,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72224689, #72312221, #72312223 |
| maximal normal subgroup of | #72224689, #72312221, #72312223 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 284349);