Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,3)(5,6,8,7)(9,10,15,16,14,20,13,19)(11,17,24,31,23,32,21,29)(12,25,27,18,28,34,26,33)(22,36,35,30,38,39,37,40);(1,2,4,3)(5,7,8,6)(10,16)(11,21)(13,15)(18,25)(19,20)(22,35)(23,24)(26,27)(29,31)(33,34)(36,39)(37,38);(9,11,14,23)(10,17,20,32)(12,22,28,38)(13,24,15,21)(16,29,19,31)(18,30,33,40)(25,39,34,36)(26,35,27,37);(1,3,4,2)(9,12,14,28)(10,18,20,33)(11,22,23,38)(13,27,15,26)(16,25,19,34)(17,30,32,40)(21,35,24,37)(29,39,31,36)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=i,d^2=efgi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279932);