Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1301 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10,14,12)(11,17,19,15)(13,20,21,16)(18,22,24,23)(25,26,30,36)(27,37,41,32)(28,43,45,34)(29,35,31,47)(33,46,52,44)(38,53,39,49)(40,54,42,50)(48,55,51,56);(1,3,4,8)(2,5,6,7)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27)(26,32)(28,39)(29,40)(30,41)(31,42)(33,51)(34,49)(35,50)(36,37)(38,45)(43,53)(44,56)(46,55)(47,54)(48,52);(3,8)(5,7)(9,12,14,10)(11,17,19,15)(13,16,21,20)(18,22,24,23)(25,28)(26,33)(27,38)(29,44)(30,45)(31,46)(32,48)(34,35)(36,52)(37,51)(39,41)(40,55)(42,56)(43,47)(49,54)(50,53);(2,6)(5,7)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,28)(26,34)(27,39)(29,46)(30,45)(31,44)(32,49)(33,35)(36,43)(37,53)(38,41)(40,55)(42,56)(47,52)(48,54)(50,51);(11,19)(15,17)(18,24)(22,23)(25,29,30,31)(26,35,36,47)(27,40,41,42)(28,44,45,46)(32,50,37,54)(33,34,52,43)(38,55,39,56)(48,53,51,49)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=fh,d^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,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, 6861454);