Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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 | #30722 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6,10,14)(7,15,17,11)(8,12,9,13)(16,20,18,19)(21,22,26,30)(23,31,34,27)(24,35,38,42)(25,39,40,29)(28,36,44,32)(33,41,47,45)(37,50,51,48)(43,46,52,49);(1,2)(3,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23,26,34)(22,27,30,31)(24,36,38,32)(25,33,40,47)(28,42,44,35)(29,41,39,45)(37,49,51,46)(43,50,52,48);(2,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,24,23,32,26,38,34,36)(22,28,27,35,30,44,31,42)(25,37,33,46,40,51,47,49)(29,43,41,48,39,52,45,50);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(21,25)(22,29)(23,33)(24,37)(26,40)(27,41)(28,43)(30,39)(31,45)(32,46)(34,47)(35,48)(36,49)(38,51)(42,50)(44,52);(2,4)(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(25,40)(29,39)(33,47)(37,51)(41,45)(43,52)(46,49)(48,50)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6594865);