Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1327 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)(3,8)(4,6)(5,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,34)(27,31,36,43)(28,32,40,47)(29,33,41,48)(35,42,49,53)(37,44,39,46)(38,45,51,55)(50,54,52,56);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(27,35)(28,38)(29,41)(31,42)(32,45)(33,48)(36,49)(37,39)(40,51)(43,53)(44,46)(47,55);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,27,29,35,30,36,41,49)(26,31,33,42,34,43,48,53)(28,39,38,52,40,37,51,50)(32,46,45,56,47,44,55,54);(10,16)(14,17)(15,22)(21,23)(27,36)(31,43)(35,49)(37,39)(42,53)(44,46)(50,52)(54,56);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(21,23)(25,28,29,38,30,40,41,51)(26,32,33,45,34,47,48,55)(27,37,35,50,36,39,49,52)(31,44,42,54,43,46,53,56)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fgh,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6743233);