Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624888 |
| 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 | |
1326 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)(13,20)(14,26)(16,28)(17,29)(18,25)(21,31)(22,32)(23,38)(24,35)(27,36)(30,40)(33,34)(37,39);(1,3)(2,5)(4,8)(6,7)(9,10,15,12)(11,16,22,21)(13,17,27,24)(14,18,26,25)(19,28,32,31)(20,29,36,35)(23,30,33,37)(34,39,38,40);(9,11,20,34,14,23,27,32,15,22,36,38,26,33,13,19)(10,16,29,39,18,30,24,31,12,21,35,40,25,37,17,28);(3,8)(5,7)(9,12,15,10)(11,21,22,16)(13,24,27,17)(14,25,26,18)(19,31,32,28)(20,35,36,29)(23,37,33,30)(34,40,38,39);(2,6)(5,7)(10,12)(11,22)(17,24)(18,25)(19,32)(23,33)(29,35)(34,38)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^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=df,e^b=ei,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72318881, #72320335, #72320368, … |
| maximal normal subgroup of | #72318881, #72320335, #72320368, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532092);