Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342800 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3489 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4120 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)(3,7)(4,6)(5,8)(9,10,14,20,15,19,13,18)(11,16,22,30,23,31,27,33)(12,24,25,34,26,17,28,32)(21,35,37,39,38,29,36,40);(1,3)(2,5)(4,8)(6,7)(9,11,15,23)(10,16,19,31)(12,21,26,38)(13,27,14,22)(17,29,24,35)(18,33,20,30)(25,37,28,36)(32,40,34,39);(9,12)(10,17)(11,21)(13,25)(14,28)(15,26)(16,29)(18,32)(19,24)(20,34)(22,36)(23,38)(27,37)(30,39)(31,35)(33,40);(3,8)(5,7);(2,6)(5,7)(9,13,15,14)(10,18,19,20)(11,22,23,27)(12,25,26,28)(16,30,31,33)(17,32,24,34)(21,36,38,37)(29,39,35,40)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72303360 |
| maximal normal subgroup of | #72303360 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6293191);