Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24404 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1497 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,17,12,16,15,23)(11,14,19,22,20,18,21,24);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,20)(13,19)(15,21)(16,18)(17,22)(23,24);(1,3,5,4)(2,7,8,6)(9,10,15,23,12,16,13,17)(11,18,19,24,20,14,21,22);(1,4,5,3)(2,7,8,6)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=efh,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72302273, #72302278 |
| maximal normal subgroup of | #72302273, #72302278 |
| maximal quotient of | #72302336 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281802);