Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623626 |
| 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 | |
1349 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)(6,8)(9,10)(11,23)(12,17)(13,22)(14,20)(15,30)(16,19)(18,27)(21,32)(24,34)(25,37)(26,38)(28,39)(29,33)(31,35)(36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,14)(13,26)(15,25)(16,24)(18,30)(19,38)(20,23)(21,35)(22,34)(27,29)(28,31)(32,40)(33,37)(36,39);(9,12,14,11)(10,18,20,37)(13,26,16,24)(15,31,29,40)(17,33,23,30)(19,36,22,35)(21,34,39,38)(25,32,27,28);(2,5)(6,8)(9,13,14,16)(10,19,20,22)(11,24,12,26)(15,28,29,32)(17,34,23,38)(18,36,37,35)(21,33,39,30)(25,40,27,31);(1,3,4,7)(2,6,5,8)(9,13,14,16)(10,19,20,22)(11,24,12,26)(15,28,29,32)(17,34,23,38)(18,36,37,35)(21,33,39,30)(25,40,27,31)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216688, #72313684, #72315876, … |
| maximal normal subgroup of | #72216688, #72313684, #72315876, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531266);