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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,30,35)(27,31,36,44)(28,33,39,47)(29,32,41,34)(37,43,49,54)(38,45,50,42)(40,46,52,48)(51,55,56,53);(3,4)(6,7)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27,30,36)(26,31,35,44)(28,37,39,49)(29,38,41,50)(32,45,34,42)(33,43,47,54)(40,51,52,56)(46,55,48,53);(1,3,5,4)(2,6,8,7)(26,32)(27,36)(29,41)(31,42)(33,46)(34,35)(37,49)(40,52)(43,53)(44,45)(47,48)(54,55);(10,15)(14,17)(16,23)(22,24)(25,28,30,39)(26,33,35,47)(27,37,36,49)(29,40,41,52)(31,43,44,54)(32,46,34,48)(38,51,50,56)(42,53,45,55);(1,4,5,3)(2,7,8,6)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(27,36)(28,39)(31,44)(33,47)(38,50)(40,52)(42,45)(46,48)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=hi,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6755472);