Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #185255 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2556 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)(3,7,8,5)(4,6)(9,10,13,18,14,16,12,17)(11,19,21,15,22,24,20,23);(1,3)(2,5)(4,8)(6,7)(9,11,14,22)(10,15,16,23)(12,21,13,20)(17,24,18,19);(1,2)(3,5)(4,6)(7,8);(10,16)(15,23)(17,18)(19,24)Polycyclic presentation:
2^9|a^2=g,b^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,f^b=fh,g^b=ghi,e^c=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72238181, #72238188, #72339520, … |
| maximal normal subgroup of | #72238181, #72238188, #72339520, … |
| maximal quotient of | #72240203, #72240230, #72240295, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92537);