Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #185475 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
3330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6,5,7,3,8)(9,10,13,18,14,17,21,24)(11,16,19,23,12,15,20,22);(2,6)(3,4)(7,8)(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);(2,7)(6,8)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,g^b=ghi,g^c=gh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72302348, #72302444, #72302458, … |
| maximal normal subgroup of | #72302348, #72302444, #72302458, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92954);