Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624128 |
| 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 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1568 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)(9,10,15,17)(11,20,21,37)(12,18,14,29)(13,25,28,39)(16,31,32,23)(19,24,35,27)(22,38,30,36)(26,33,40,34);(1,3)(2,5)(4,8)(6,7)(11,21)(16,32)(20,37)(22,30)(23,31)(26,40)(33,34)(36,38);(9,11)(10,16)(12,22)(13,26)(14,30)(15,21)(17,32)(18,33)(19,36)(20,25)(23,27)(24,31)(28,40)(29,34)(35,38)(37,39);(3,8)(5,7)(10,17)(16,32)(18,29)(19,35)(20,37)(25,39)(33,34)(36,38);(2,6)(5,7)(9,12,15,14)(10,18,17,29)(11,22,21,30)(13,24,28,27)(16,33,32,34)(19,25,35,39)(20,38,37,36)(23,26,31,40)Polycyclic presentation:
2^9|a^2=i,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df,g^c=gh,h^c=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72314141, #72315864, #72315871, … |
| maximal normal subgroup of | #72314141, #72315864, #72315871, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531345);