Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632082 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26,31,36)(27,32,37,47)(28,34,42,50)(29,35,30,33)(38,45,51,55)(39,44,40,46)(41,48,43,49)(52,56,53,54);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,18,24)(12,20,13,19)(16,23,17,22)(25,27)(26,32)(28,38)(29,39)(30,40)(31,37)(33,46)(34,45)(35,44)(36,47)(41,52)(42,51)(43,53)(48,56)(49,54)(50,55);(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,24,23)(26,33)(27,37)(29,30)(32,44)(34,48)(35,36)(38,51)(41,43)(45,54)(46,47)(49,50)(55,56);(2,5)(6,7)(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,23,24,22)(25,28,31,42)(26,34,36,50)(27,38,37,51)(29,43,30,41)(32,45,47,55)(33,48,35,49)(39,53,40,52)(44,54,46,56);(1,3,4,8)(2,6,5,7)(10,18)(15,24)(16,17)(22,23)(25,29,31,30)(26,33,36,35)(27,39,37,40)(28,41,42,43)(32,46,47,44)(34,49,50,48)(38,52,51,53)(45,54,55,56)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,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^c=ei,g^c=gi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6559757);