Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1194 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)(3,7)(4,8)(5,6)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26,31,36)(27,32,40,47)(28,33,42,49)(29,34,30,35)(37,44,39,46)(38,45,52,55)(41,48,43,50)(51,54,53,56);(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27,31,40)(26,32,36,47)(28,38,42,52)(29,39,30,37)(33,45,49,55)(34,46,35,44)(41,51,43,53)(48,54,50,56);(2,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(27,37)(29,30)(32,44)(34,35)(38,51)(39,40)(41,43)(45,54)(46,47)(48,50)(52,53)(55,56);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,28)(26,33)(27,38)(29,43)(30,41)(31,42)(32,45)(34,50)(35,48)(36,49)(37,51)(39,53)(40,52)(44,54)(46,56)(47,55);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,29)(26,34)(28,41)(30,31)(33,48)(35,36)(37,39)(38,52)(42,43)(44,46)(45,55)(49,50)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=f,d^2=fg,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6652120);