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 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1335 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)(12,20)(13,23)(15,25)(17,26)(18,30)(21,31)(22,35)(24,34)(27,36)(28,40)(29,39)(32,33)(37,38);(1,3)(2,5)(4,8)(6,7)(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);(3,8)(5,7)(11,21)(15,27)(19,31)(22,32)(25,36)(28,37)(33,35)(38,40);(2,6)(5,7)(10,16)(15,27)(17,29)(18,30)(25,36)(26,39)(28,37)(38,40)Polycyclic presentation:
2^9|b^2=i,c^2=gh,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=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72287059, #72287344, #72287357, … |
| maximal normal subgroup of | #72287059, #72287344, #72287357, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532039);