Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624888 |
| 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 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(11,19)(12,20)(13,23)(15,25)(17,26)(18,30)(21,31)(22,35)(24,34)(27,36)(28,40)(29,39)(32,33)(37,38);(1,2)(3,6)(4,5)(7,8)(9,10,14,16)(11,15,21,27)(12,17,24,29)(13,18,23,30)(19,25,31,36)(20,26,34,39)(22,28,32,37)(33,38,35,40);(9,11,20,33,13,22,24,31,14,21,34,35,23,32,12,19)(10,15,26,38,18,28,29,36,16,27,39,40,30,37,17,25);(2,5)(6,8)(11,21)(15,27)(19,31)(22,32)(25,36)(28,37)(33,35)(38,40);(1,3,4,7)(2,6,5,8)(10,16)(11,21)(17,29)(18,30)(19,31)(22,32)(26,39)(33,35)Polycyclic presentation:
2^9|b^2=i,c^2=gh,e^2=f,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^b=ei,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72287059, #72314019, #72315837, … |
| maximal normal subgroup of | #72287059, #72314019, #72315837, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532220);