Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624150 |
| 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 ◁ #532 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1584 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)(11,18)(12,22)(13,20)(14,29)(15,19)(16,24)(17,25)(21,31)(23,33)(26,36)(27,38)(28,32)(30,37)(34,40)(35,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,26)(13,24)(14,25)(15,23)(17,32)(18,20)(19,36)(21,34)(22,33)(27,40)(28,39)(29,35)(30,31)(37,38);(9,11)(10,17)(12,23)(13,24)(14,30)(15,26)(16,32)(18,29)(19,34)(20,35)(21,36)(22,37)(25,31)(27,39)(28,40)(33,38);(3,8)(5,7)(9,11)(10,18)(12,26)(13,24)(14,25)(15,23)(16,20)(17,29)(19,33)(21,37)(22,36)(27,40)(28,39)(30,31)(32,35)(34,38);(2,6)(5,7)(9,12,13,15)(10,19,20,22)(11,23,24,26)(14,27,28,31)(16,33,18,36)(17,34,35,37)(21,32,38,29)(25,30,39,40)Polycyclic presentation:
2^9|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=dfi,e^b=ei,g^c=gh,h^c=hi,e^d=ei71 conjugacy classes — view the full character table.
| maximal subgroup of | #72315208, #72315837, #72315876, … |
| maximal normal subgroup of | #72315208, #72315837, #72315876, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531425);