Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,22,24,16)(13,17)(14,18)(19,31,32,26)(20,27)(21,28)(23,35,36,29)(25,30)(33,39,40,37)(34,38);(2,5)(6,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,26)(17,28)(18,27)(22,31)(23,33)(24,32)(25,34)(29,37)(30,38)(35,39)(36,40);(9,12)(10,16)(11,19)(13,23)(14,24)(15,26)(17,29)(18,22)(20,32)(21,33)(25,36)(27,31)(28,37)(30,35)(34,40)(38,39);(1,3,4,7)(2,6,5,8)(11,20)(15,27)(19,32)(21,34)(26,31)(28,38)(33,40)(37,39);(9,13,14,25)(10,17,18,30)(11,21,20,34)(12,23,24,36)(15,28,27,38)(16,29,22,35)(19,33,32,40)(26,37,31,39)Polycyclic presentation:
2^9|a^2=h,d^2=f,e^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=di,h^c=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72303173, #72303551, #72303560, … |
| maximal normal subgroup of | #72303173, #72303551, #72303560, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278136);