Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #185475 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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 | D4 |
| 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:
(1,2,3,6,4,5,7,8)(9,10,13,18,14,16,12,17)(11,19,21,15,22,24,20,23);(2,5)(3,7)(9,11,14,22)(10,15,16,23)(12,21,13,20)(17,24,18,19);(2,5)(3,7);(10,16)(15,23)(17,18)(19,24)Polycyclic presentation:
2^9|a^2=g,b^2=i,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 | #72238937, #72238971, #72257616, … |
| maximal normal subgroup of | #72238937, #72238971, #72257616, … |
| maximal quotient of | #72264409 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92956);