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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1262 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)(25,26,29,34,30,35,42,32)(27,31,38,43,36,45,51,46)(28,33,41,50,40,49,54,47)(37,44,52,55,39,48,53,56);(1,2)(3,7)(4,5)(6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27)(26,31)(28,39)(29,38)(30,36)(32,46)(33,48)(34,43)(35,45)(37,40)(41,53)(42,51)(44,49)(47,55)(50,56)(52,54);(2,5)(6,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,32)(27,36)(29,42)(31,43)(33,47)(34,35)(37,39)(41,54)(44,55)(45,46)(48,56)(49,50);(10,16)(14,17)(15,22)(21,23)(25,28,30,40)(26,33,35,49)(27,37,36,39)(29,41,42,54)(31,44,45,48)(32,47,34,50)(38,52,51,53)(43,55,46,56);(1,3,4,8)(2,6,5,7)(11,19)(14,17)(18,24)(21,23)(27,36)(28,40)(31,45)(33,49)(38,51)(41,54)(43,46)(47,50)Polycyclic presentation:
2^9|a^2=g,c^2=f,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,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, 6701367);