Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1362 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)(3,8)(4,6)(5,7)(9,10,14,18,15,20,13,19)(11,16,23,31,24,32,22,36)(12,25,27,17,28,34,29,35)(21,33,37,30,38,40,26,39);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,26)(13,22)(14,23)(15,24)(17,33)(18,31)(19,36)(20,32)(21,27)(25,39)(28,37)(29,38)(30,34)(35,40);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,21)(13,29)(14,27)(15,28)(16,30)(18,34)(19,25)(20,35)(22,26)(23,37)(24,38)(31,40)(32,39)(33,36);(2,7)(6,8)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267150);