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 | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1677 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)(23,27)(24,35)(25,29)(26,31)(28,32)(30,34)(33,40)(36,48)(37,39)(38,43)(41,45)(42,46)(44,47)(49,50)(51,52);(2,3)(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23,26,34)(22,27,31,30)(24,32,37,46)(25,33,38,47)(28,39,42,35)(29,40,43,44)(36,45,49,52)(41,50,51,48);(1,2,4,3)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,24,34,46,26,37,23,32)(22,28,27,39,31,42,30,35)(25,36,47,52,38,49,33,45)(29,41,40,50,43,51,44,48);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(21,25,26,38)(22,29,31,43)(23,33,34,47)(24,36,37,49)(27,40,30,44)(28,41,42,51)(32,45,46,52)(35,48,39,50);(1,3,4,2)(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(25,38)(29,43)(33,47)(36,49)(40,44)(41,51)(45,52)(48,50)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fghi,d^2=fi,e^2=fh,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, 6724739);