Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2002 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,12)(7,14)(8,16)(9,10)(13,20)(15,17);(1,2,3,4)(5,6,9,12,11,14,10,7)(8,13,19,20,16,17,18,15);(2,4)(5,7,9,6,11,12,10,14)(8,17,18,20,16,13,19,15);(5,8,10,18,11,16,9,19)(6,13,7,15,14,17,12,20)Polycyclic presentation:
2^9|b^2=fg,c^2=e,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,h^b=hi,d^c=de,h^c=hi,e^d=ei,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72338485, #72338488, #72338490 |
| maximal normal subgroup of | #72338485, #72338488, #72338490 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 373250);