Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3318 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,8)(6,7)(9,10)(11,14)(12,19,20,15)(13,16)(17,23,24,21)(18,22);(1,2,4,6)(3,7,8,5)(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23);(2,6)(5,7)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23);(2,7)(3,8)(5,6)Polycyclic presentation:
2^9|a^2=g,b^2=hi,e^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=de,e^b=eh,g^c=gi,e^d=eh65 conjugacy classes — view the full character table.
| maximal subgroup of | #72268288, #72268291, #72268311, … |
| maximal normal subgroup of | #72268288, #72268291, #72268311, … |
| maximal quotient of | #72313349, #72313663 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 127465);