Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624876 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | #62251 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,20)(12,17)(13,26)(14,29)(15,19)(16,23)(18,27)(21,34)(22,35)(24,37)(25,36)(28,30)(31,39)(32,38)(33,40);(1,3)(2,5)(4,8)(6,7);(9,11,21,38,14,22,13,23,15,24,27,39,12,25,28,40)(10,16,30,35,17,32,18,20,19,33,26,36,29,31,34,37);(3,8)(5,7);(2,6)(5,7)(9,12,15,14)(10,17,19,29)(11,22,24,25)(13,21,28,27)(16,31,33,32)(18,26,34,30)(20,35,37,36)(23,39,40,38)Polycyclic presentation:
2^9|c^2=ghi,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72314396, #72315814, #72315916 |
| maximal normal subgroup of | #72314396, #72315814, #72315916 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532077);