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×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,14,17)(11,18,20,15)(12,21,13,16)(19,23,22,24)(25,26,30,35,29,34,43,32)(27,36,39,45,38,31,44,47)(28,40,42,48,41,33,49,50)(37,46,53,56,52,51,54,55);(1,2,4,5)(3,7,8,6)(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(25,27)(26,31)(28,37)(29,38)(30,39)(32,45)(33,46)(34,36)(35,47)(40,51)(41,52)(42,53)(43,44)(48,55)(49,54)(50,56);(2,5)(6,7)(26,32)(30,43)(31,45)(33,48)(34,35)(36,47)(39,44)(40,50)(42,49)(46,55)(51,56)(53,54);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,28,29,41)(26,33,34,40)(27,37,38,52)(30,42,43,49)(31,46,36,51)(32,48,35,50)(39,53,44,54)(45,55,47,56);(1,3,4,8)(2,6,5,7)(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(27,38)(28,41)(31,36)(33,40)(39,44)(42,49)(45,47)(48,50)Polycyclic presentation:
2^9|a^2=fgi,b^2=fhi,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,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, 6711005);