Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624842 |
| 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 | |
1333 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)(11,19)(12,21)(13,24)(15,26)(17,27)(18,30)(20,31)(22,38)(23,34)(25,37)(28,40)(29,35)(32,33)(36,39);(1,3)(2,5)(4,8)(6,7)(9,10)(11,20)(12,17)(13,18)(14,16)(15,23)(19,31)(21,27)(22,36)(24,30)(25,29)(26,34)(28,32)(33,40)(35,37)(38,39);(9,11,21,33,13,22,25,34,14,23,37,38,24,32,12,19)(10,15,27,39,18,28,29,31,16,20,35,40,30,36,17,26);(3,8)(5,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,26)(20,23)(21,27)(22,28)(24,30)(25,29)(31,34)(32,36)(33,39)(35,37)(38,40);(2,6)(5,7)(10,16)(15,20)(17,29)(18,30)(26,31)(27,35)(28,36)(39,40)Polycyclic presentation:
2^9|c^2=gh,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72287098, #72287240, #72287254, … |
| maximal normal subgroup of | #72287098, #72287240, #72287254, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532060);