Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535815 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1270 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)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26)(27,36)(28,33)(29,34)(30,44)(31,35)(32,39)(37,50)(38,47)(40,48)(41,49)(42,54)(43,46)(45,51)(52,55)(53,56);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,30,39,31,38,29,43)(26,32,34,36,35,46,44,47)(28,37,42,51,40,52,41,53)(33,45,49,50,48,56,54,55);(2,6)(7,8)(11,17)(14,20)(18,23)(21,24)(25,28)(26,33)(27,37)(29,41)(30,42)(31,40)(32,45)(34,49)(35,48)(36,50)(38,52)(39,51)(43,53)(44,54)(46,56)(47,55);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(27,38)(28,40)(32,46)(33,48)(36,47)(39,43)(41,42)(49,54);(1,4,5,3)(2,8,6,7)(12,19)(15,22)(18,23)(21,24)(25,29,31,30)(26,34,35,44)(27,39,38,43)(28,41,40,42)(32,47,46,36)(33,49,48,54)(37,51,52,53)(45,55,56,50)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=g,e^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6472884);