Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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)(27,33)(28,34)(29,35)(30,45)(31,37)(32,38)(36,46)(39,47)(40,48)(41,49)(42,50)(43,52)(44,53)(51,55)(54,56);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27,32,40)(26,33,38,48)(28,41,44,55)(29,30,31,46)(34,49,53,51)(35,45,37,36)(39,42,54,43)(47,50,56,52);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,31,43,32,44,29,42)(26,34,37,52,38,53,35,50)(27,39,30,41,40,54,46,55)(33,47,45,49,48,56,36,51);(10,16)(14,17)(15,22)(21,23)(25,29,32,31)(26,35,38,37)(27,30,40,46)(28,42,44,43)(33,45,48,36)(34,50,53,52)(39,41,54,55)(47,49,56,51);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(21,23)(25,30,31,40,32,46,29,27)(26,36,37,33,38,45,35,48)(28,41,43,54,44,55,42,39)(34,51,52,47,53,49,50,56)Polycyclic presentation:
2^9|b^2=fi,c^2=fgh,d^2=i,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6743067);