Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1231 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26)(27,37)(28,41)(29,35)(30,36)(31,33)(32,38)(34,43)(39,47)(40,46)(42,50)(44,48)(45,49)(51,52)(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)(26,32)(28,39)(29,44)(30,38)(31,40)(33,48)(34,47)(35,46)(36,37)(41,51)(42,54)(43,52)(45,53)(49,56)(50,55);(2,6)(7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,33)(27,38)(29,31)(32,46)(34,49)(35,36)(37,48)(39,52)(41,50)(42,45)(47,55)(51,56);(1,3,5,4)(2,7,6,8)(10,16)(14,17)(15,22)(21,23)(25,28)(26,34)(27,39)(29,45)(30,43)(31,42)(32,47)(33,49)(35,50)(36,41)(37,51)(38,52)(40,54)(44,53)(46,55)(48,56);(1,4,5,3)(2,7,6,8)(11,19)(14,17)(18,24)(21,23)(25,29,30,31)(26,35,36,33)(27,40,38,44)(28,42,43,45)(32,48,37,46)(34,49,41,50)(39,53,52,54)(47,55,51,56)Polycyclic presentation:
2^9|a^2=fi,c^2=fi,d^2=hi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6767611);