Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | |
1396 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)(27,36)(28,40)(29,35)(30,32)(31,45)(33,39)(34,42)(37,48)(38,53)(41,50)(43,46)(44,49)(47,54)(51,52)(55,56);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(26,32)(27,37)(34,40)(35,45)(38,52)(39,43)(49,50)(54,55);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,27,31,39,30,37,29,43)(26,33,35,36,32,46,45,48)(28,38,41,55,42,52,44,54)(34,47,50,51,40,56,49,53);(3,8)(5,7)(10,15)(14,17)(16,23)(22,24)(25,28)(26,34)(27,38)(29,44)(30,42)(31,41)(32,40)(33,47)(35,50)(36,51)(37,52)(39,55)(43,54)(45,49)(46,56)(48,53);(2,6)(5,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,29,30,31)(26,35,32,45)(27,39,37,43)(28,41,42,44)(33,48,46,36)(34,49,40,50)(38,54,52,55)(47,51,56,53)Polycyclic presentation:
2^9|b^2=fi,c^2=g,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,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6820109);