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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1371 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)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26)(27,37)(28,41)(29,35)(30,36)(31,33)(32,40)(34,42)(38,47)(39,48)(43,50)(44,46)(45,49)(51,52)(53,56)(54,55);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27)(26,32)(28,38)(29,44)(30,40)(31,39)(33,46)(34,47)(35,48)(36,37)(41,51)(42,52)(43,54)(45,53)(49,55)(50,56);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(26,33)(28,42)(29,31)(32,46)(34,50)(35,36)(37,48)(38,52)(39,44)(41,49)(47,56)(51,55);(3,8)(5,7)(10,15)(14,17)(16,23)(22,24)(25,28)(26,34)(27,38)(29,45)(30,42)(31,43)(32,47)(33,49)(35,50)(36,41)(37,51)(39,54)(40,52)(44,53)(46,55)(48,56);(2,6)(5,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,29,30,31)(26,35,36,33)(27,39,40,44)(28,43,42,45)(32,46,37,48)(34,49,41,50)(38,53,52,54)(47,56,51,55)Polycyclic presentation:
2^9|b^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^c=ei,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, 6796973);