Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1779 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)(7,15)(8,12)(9,14)(10,13)(11,17)(16,20)(18,19)(21,22,27,31)(23,32,35,42)(24,29,36,44)(25,30,26,39)(28,40,43,34)(33,47,48,52)(37,45,38,46)(41,49,51,50);(1,2)(3,4)(5,7,10,17)(6,11,13,15)(8,16,9,18)(12,19,14,20)(24,36)(29,44)(33,48)(37,38)(41,51)(45,46)(47,52)(49,50);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(21,23)(22,28)(24,33)(25,34)(26,40)(27,35)(29,41)(30,42)(31,43)(32,39)(36,48)(37,49)(38,50)(44,51)(45,47)(46,52);(6,13)(11,15)(12,14)(19,20)(21,24,27,36)(22,29,31,44)(23,33,35,48)(25,38,26,37)(28,41,43,51)(30,46,39,45)(32,47,42,52)(34,50,40,49);(2,4)(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(21,25,27,26)(22,30,31,39)(23,34,35,40)(24,37,36,38)(28,42,43,32)(29,45,44,46)(33,49,48,50)(41,47,51,52)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6582355);