Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1332 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)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26)(27,37)(28,33)(29,35)(30,34)(31,36)(32,39)(38,52)(40,48)(41,49)(42,50)(43,51)(44,46)(45,54)(47,55)(53,56);(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,30,39)(26,32,34,37)(28,40,41,55)(29,31,44,43)(33,47,49,48)(35,51,46,36)(38,53,54,42)(45,56,52,50);(2,5)(6,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,31,42,30,41,43,53)(26,33,36,50,34,49,51,56)(27,38,29,40,39,54,44,55)(32,45,46,47,37,52,35,48);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(26,34)(28,41)(32,37)(35,46)(36,51)(38,54)(40,55)(42,53);(1,3,4,7)(2,6,5,8)(11,21)(15,18)(19,20)(23,24)(25,29,43,27,30,44,31,39)(26,35,51,37,34,46,36,32)(28,40,53,38,41,55,42,54)(33,48,56,52,49,47,50,45)Polycyclic presentation:
2^9|b^2=i,c^2=fhi,d^2=fi,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6738656);