Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #186835 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3486 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,19,22,33)(12,16)(13,18)(14,17)(15,21,29,25)(20,36,38,35)(23,31)(24,30)(26,34)(27,32)(28,37,40,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,20)(13,25)(14,22)(16,28)(17,29)(18,33)(19,32)(21,27)(23,39)(24,38)(26,37)(30,40)(31,35)(34,36);(9,12,14,24)(10,16,17,30)(11,20,22,38)(13,26,27,23)(15,28,29,40)(18,34,32,31)(19,35,33,36)(21,39,25,37);(10,17)(15,29)(16,30)(18,32)(19,33)(28,40)(31,34)(35,36)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280989 |
| maximal normal subgroup of | #72280989 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94034);