Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7540 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1415 |
| commutator | C32 | C42 |
| Frattini, cosocle | C22×C32 | C22 |
| prime radical | C23 | D32 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 68 points:
(1,2,3,4)(5,6,9,15)(7,20,22,42)(8,26,29,16)(10,14,31,43)(11,38,32,57)(12,27,33,21)(13,41,34,19)(17,51,46,28)(18,53,47,39)(23,40,56,54)(24,58,45,37)(25,59,49,35)(30,50,55,62)(36,61,60,48)(44,67,66,64)(52,63,65,68);(5,7,10,23,11,24,35,19,39,50,64,52,63,44,30,47,13,25,37,57,40,14,42,15,33,26,60,46,51,61,29,21,9,22,31,56,32,45,59,41,53,62,67,65,68,66,55,18,34,49,58,38,54,43,20,6,12,16,36,17,28,48,8,27)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fg,e^2=f,f^2=gh,g^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fg,g^a=gh,h^a=hi,c^b=ci92 conjugacy classes — view the full character table.
| maximal subgroup of | #72286491, #72286843, #72286857, … |
| maximal normal subgroup of | #72286491, #72286843, #72286857, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1963);