Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1173 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,13,15)(11,17,19,14)(12,16,21,23)(18,24,20,22)(25,26)(27,33)(28,43)(29,35)(30,36)(31,46)(32,37)(34,44)(38,53)(39,48)(40,49)(41,56)(42,50)(45,52)(47,54)(51,55);(1,2)(3,6)(4,5)(7,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27,32,42)(26,33,37,50)(28,39,29,38)(30,40,31,41)(34,51,52,47)(35,53,43,48)(36,49,46,56)(44,55,45,54);(2,5)(6,8)(25,28,32,29)(26,34,37,52)(27,38,42,39)(30,44,31,45)(33,47,50,51)(35,46,43,36)(40,54,41,55)(48,56,53,49);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,23)(22,24)(25,29,32,28)(26,35,37,43)(27,39,42,38)(30,44,31,45)(33,48,50,53)(34,46,52,36)(40,54,41,55)(47,56,51,49);(2,5)(6,8)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,30,32,31)(26,36,37,46)(27,40,42,41)(28,44,29,45)(33,49,50,56)(34,35,52,43)(38,54,39,55)(47,48,51,53)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=i,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,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, 6683821);