Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196248 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1767 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)(4,5)(6,8)(9,10,13,17,14,16,12,22)(11,18,21,23,20,24,19,15);(2,5)(6,7)(9,11)(10,15,16,23)(12,19)(13,21)(14,20)(17,18,22,24);(1,3,4,8)(2,6,5,7);(10,16)(12,13)(15,23)(19,21)Polycyclic presentation:
2^9|a^2=h,b^2=efh,c^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,g^b=gi,e^d=ei,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239300, #72257517, #72257542 |
| maximal normal subgroup of | #72239300, #72257517, #72257542 |
| maximal quotient of | #72239307, #72257954 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300544);