Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11796 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C2×#2397 | C2 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,19,21,24)(12,17)(13,18)(14,16)(15,20,23,22);(2,5)(3,8)(6,7)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24);(2,6)(5,7)(10,16)(15,23)(17,18)(19,24);(9,12,14,13)(10,17,16,18)(11,20,21,22)(15,19,23,24)Polycyclic presentation:
2^9|a^2=f,d^2=hi,e^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231142, #72231150, #72231173, … |
| maximal normal subgroup of | #72231142, #72231150, #72231173, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389146);