Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 | #2535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1259 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,37)(28,42)(29,35)(30,36)(31,33)(32,40)(34,45)(38,48)(39,47)(41,49)(43,51)(44,52)(46,50)(53,55)(54,56);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,30,40)(26,32,36,37)(28,43,45,38)(29,39,31,41)(33,47,35,49)(34,51,42,48)(44,54,46,53)(50,56,52,55);(2,6)(7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,33)(29,31)(32,47)(34,50)(35,36)(37,49)(39,41)(42,52)(44,46)(48,55)(51,56)(53,54);(1,3,5,4)(2,7,6,8)(10,16)(14,17)(15,22)(21,23)(25,28)(26,34)(27,38)(29,46)(30,45)(31,44)(32,48)(33,50)(35,52)(36,42)(37,51)(39,54)(40,43)(41,53)(47,55)(49,56);(1,4,5,3)(2,7,6,8)(11,19)(14,17)(18,24)(21,23)(25,29,30,31)(26,35,36,33)(27,39,40,41)(28,44,45,46)(32,49,37,47)(34,50,42,52)(38,53,43,54)(48,55,51,56)Polycyclic presentation:
2^9|b^2=i,c^2=fi,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,e^c=ei,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6767346);