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 | |
1176 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)(25,26,30,36)(27,37,38,32)(28,42,45,34)(29,35,31,33)(39,48,43,51)(40,49,41,47)(44,52,46,50)(53,56,54,55);(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)(28,43)(29,40)(30,38)(31,41)(33,49)(34,51)(35,47)(36,37)(39,45)(42,48)(44,54)(46,53)(50,55)(52,56);(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,24,23)(26,33)(27,38)(29,31)(32,47)(34,50)(35,36)(37,49)(39,43)(42,52)(44,46)(48,55)(51,56);(2,5)(6,7)(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,23,24,22)(25,28,30,45)(26,34,36,42)(27,39,38,43)(29,46,31,44)(32,48,37,51)(33,50,35,52)(40,54,41,53)(47,55,49,56);(1,3,4,8)(2,6,5,7)(10,18)(15,24)(16,17)(22,23)(25,29,30,31)(26,35,36,33)(27,40,38,41)(28,44,45,46)(32,47,37,49)(34,50,42,52)(39,53,43,54)(48,56,51,55)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=hi,d^2=h,e^2=f,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, 6559740);