Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1685 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22,27,32)(23,28,37,45)(24,30,39,47)(25,31,26,41)(29,40,48,38)(33,49,34,43)(35,44,36,52)(42,51,46,50);(2,3)(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23)(22,28)(24,34)(25,35)(26,36)(27,37)(29,46)(30,43)(31,44)(32,45)(33,39)(38,51)(40,50)(41,52)(42,48)(47,49);(1,2,4,3)(21,24)(22,29)(23,33)(25,38)(26,40)(27,39)(28,42)(30,41)(31,47)(32,48)(34,37)(35,50)(36,51)(43,44)(45,46)(49,52);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,24)(22,30)(23,34)(25,40)(26,38)(27,39)(28,43)(29,41)(31,48)(32,47)(33,37)(35,50)(36,51)(42,44)(45,49)(46,52);(1,3,4,2)(7,15)(10,13)(14,20)(18,19)(21,25,27,26)(22,31,32,41)(23,35,37,36)(24,38,39,40)(28,44,45,52)(29,47,48,30)(33,50,34,51)(42,43,46,49)Polycyclic presentation:
2^9|a^2=fi,c^2=hi,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6716159);