Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632088 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
1367 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,21)(19,22)(23,24)(25,26,31,36)(27,32,37,46)(28,34,41,50)(29,35,30,33)(38,45,51,56)(39,47,42,44)(40,48,43,49)(52,54,53,55);(3,4)(6,7)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,27,31,37)(26,32,36,46)(28,38,41,51)(29,42,30,39)(33,47,35,44)(34,45,50,56)(40,53,43,52)(48,55,49,54);(1,3,5,4)(2,6,8,7)(26,33)(27,37)(29,30)(32,44)(34,48)(35,36)(38,51)(40,43)(45,54)(46,47)(49,50)(55,56);(1,4,5,3)(2,7,8,6)(10,15)(11,18)(16,20)(19,24)(25,28,31,41)(26,34,36,50)(27,38,37,51)(29,43,30,40)(32,45,46,56)(33,48,35,49)(39,53,42,52)(44,54,47,55);(2,8)(6,7)(9,12,13,21)(10,16,15,20)(11,19,18,24)(14,22,17,23)(25,29,31,30)(26,33,36,35)(27,39,37,42)(28,40,41,43)(32,44,46,47)(34,49,50,48)(38,52,51,53)(45,55,56,54)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfhi,e^b=ei,e^c=ei,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6558257);