Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422233 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1091 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(7,8)(9,10)(11,21)(12,15)(13,27)(14,19)(16,20)(17,24)(18,30)(22,25)(23,37)(26,29)(28,40)(31,36)(32,35)(33,38)(34,39);(1,2,5,6)(3,7,4,8)(9,11,16,25,14,24,15,31)(10,17,20,36,19,21,12,22)(13,28,32,39,29,23,30,38)(18,34,27,37,35,33,26,40);(1,3,5,4)(2,7,6,8)(9,12,16,19,14,20,15,10)(11,22,31,17,24,36,25,21)(13,26,32,35,29,27,30,18)(23,37,39,33,28,40,38,34);(1,4,5,3)(2,7,6,8)(9,13)(10,18)(11,23)(12,26)(14,29)(15,30)(16,32)(17,33)(19,35)(20,27)(21,34)(22,37)(24,28)(25,38)(31,39)(36,40)Polycyclic presentation:
2^9|b^2=efg,c^2=ef,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 346522);