Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1247 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)(30,45)(31,35)(32,39)(37,52)(38,48)(40,54)(41,50)(42,51)(43,56)(44,47)(46,53)(49,55);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,29,44,31,38,30,39)(26,32,45,48,35,47,34,36)(28,40,42,53,41,37,43,55)(33,49,56,52,50,46,51,54);(2,6)(7,8)(11,17)(14,20)(18,23)(21,24)(25,28)(26,33)(27,37)(29,42)(30,43)(31,41)(32,46)(34,51)(35,50)(36,52)(38,40)(39,53)(44,55)(45,56)(47,49)(48,54);(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)(28,41)(32,47)(33,50)(36,48)(39,44)(42,43)(51,56);(1,4,5,3)(2,8,6,7)(12,19)(15,22)(18,23)(21,24)(25,29,31,30)(26,34,35,45)(27,39,38,44)(28,42,41,43)(32,48,47,36)(33,51,50,56)(37,53,40,55)(46,54,49,52)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,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=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6472912);