Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| 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 | #2619 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1257 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24)(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);(1,3,7,4)(2,5,8,6)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(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);(1,4,7,3)(2,5,8,6)(10,15)(11,17)(12,19)(20,24)(26,32)(28,41)(30,43)(31,45)(33,50)(34,35)(36,47)(37,52)(39,44)(40,48)(46,56)(51,55);(1,5,7,6)(2,3,8,4)(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24)(25,28)(26,33)(27,37)(29,41)(30,42)(31,46)(32,48)(34,40)(35,50)(36,51)(38,52)(39,53)(43,49)(44,54)(45,55)(47,56);(1,6,7,5)(2,3,8,4)(11,17)(14,21)(18,23)(20,24)(27,38)(28,41)(31,36)(33,40)(39,44)(42,49)(45,47)(48,50)Polycyclic presentation:
2^9|a^2=gh,b^2=fi,c^2=fi,d^2=fhi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,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, 6855412);