Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196092 |
| 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 | |
1707 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)(9,10,13,16,14,18,12,17)(11,19,21,23,22,24,20,15);(2,5)(6,7)(9,11)(10,15,18,23)(12,20)(13,21)(14,22)(16,19,17,24);(1,3,4,8)(2,6,5,7);(10,16)(12,13)(15,19)(17,18)(20,21)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=efh,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^b=gi,e^d=ei,g^d=gi,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72252496, #72252510, #72252518, … |
| maximal normal subgroup of | #72252496, #72252510, #72252518, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300569);