Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1635 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,4,3)(8,16)(11,17)(15,20)(18,19)(21,22,26,32,27,33,25,31)(23,28,36,44,37,45,40,49)(24,30,38,47,39,48,41,29)(34,50,52,46,35,43,51,42);(2,3)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,27,37)(22,28,33,45)(24,35,39,34)(25,40,26,36)(29,46,47,42)(30,43,48,50)(31,49,32,44)(38,51,41,52);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24,27,39)(22,29,33,47)(23,34,37,35)(25,38,26,41)(28,42,45,46)(30,32,48,31)(36,51,40,52)(43,49,50,44);(1,3,4,2)(7,14)(10,13)(15,20)(18,19)(21,24)(22,30)(23,35)(25,41)(26,38)(27,39)(28,43)(29,31)(32,47)(33,48)(34,37)(36,51)(40,52)(42,44)(45,50)(46,49);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,25,27,26)(22,31,33,32)(23,36,37,40)(24,38,39,41)(28,44,45,49)(29,30,47,48)(34,51,35,52)(42,43,46,50)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=i,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 7091009);