Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8359 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#379 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6,7,8)(9,10,13,18,14,17,21,24)(11,16,19,23,12,15,20,22);(1,2)(3,6)(4,8)(5,7)(9,11,14,12)(10,15,17,16)(13,20,21,19)(18,23,24,22);(9,12,14,11)(10,16,17,15)(13,19,21,20)(18,22,24,23);(1,3,5,4)(2,6,7,8)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=i,e^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,g^b=ghi,g^c=gh86 conjugacy classes — view the full character table.
| maximal subgroup of | #72268183, #72268189, #72268191, … |
| maximal normal subgroup of | #72268183, #72268189, #72268191, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92736);