Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459145 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2751 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16788 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
711 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,25,7,11,26,12)(3,8,16,29,15,18,17,28)(4,14,19,9,20,32,27,30)(5,21,23,31,24,10,22,13);(1,3,6,17,7,15,26,16)(2,8,12,29,11,18,25,28)(4,10,27,13,20,21,19,31)(5,14,22,9,24,32,23,30);(1,4,6,19,7,20,26,27)(2,9,12,14,11,30,25,32)(3,13,16,21,15,31,17,10)(5,18,22,29,24,8,23,28);(1,5,6,22,7,24,26,23)(2,10,25,31,11,21,12,13)(3,14,17,9,15,32,16,30)(4,18,19,29,20,8,27,28)Polycyclic presentation:
2^9|a^2=eg,b^2=efh,c^2=f,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=de,h^c=hi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72319935, #72319962, #72320075, … |
| maximal normal subgroup of | #72319935, #72319962, #72320075, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378536);