Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200022 |
| 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 | #53064 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1447 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)(3,5)(4,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,32)(28,34)(29,35)(30,33)(31,36)(37,45)(38,44)(39,47)(40,48)(41,50)(42,46)(43,49)(51,55)(52,54)(53,56);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27)(26,32)(28,37)(29,42)(30,38)(31,39)(33,44)(34,45)(35,46)(36,47)(40,52)(41,53)(43,51)(48,54)(49,55)(50,56);(3,7)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,33)(29,30)(32,44)(34,48)(35,36)(38,42)(40,43)(45,54)(46,47)(49,50)(51,52)(55,56);(2,6)(3,7)(11,19)(14,17)(18,24)(21,23)(25,28,31,41)(26,34,36,50)(27,37,39,53)(29,43,30,40)(32,45,47,56)(33,48,35,49)(38,52,42,51)(44,54,46,55);(3,7)(5,8)(10,16)(14,17)(15,22)(21,23)(25,29,31,30)(26,35,36,33)(27,38,39,42)(28,40,41,43)(32,44,47,46)(34,48,50,49)(37,51,53,52)(45,55,56,54)Polycyclic presentation:
2^9|c^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7107821);