Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196290 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1662 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)(9,10,12,17,14,18,13,16)(11,15,19,24,21,22,20,23);(2,5)(6,8)(9,11,12,19,14,21,13,20)(10,15,17,24,18,22,16,23);(1,3,4,7)(2,6,5,8)(10,16)(11,19)(12,13)(15,22)(17,18)(20,21);(2,5)(6,8)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72310607, #72310633, #72310636, … |
| maximal normal subgroup of | #72310607, #72310633, #72310636, … |
| maximal quotient of | C2×#364697, #72316755, #72316996 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271893);