Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632074 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1179 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(3,7)(6,8)(9,10,14,17)(11,18,20,15)(12,16,13,22)(19,24,21,23)(25,26,30,34)(27,31,36,44)(28,33,39,48)(29,32,41,35)(37,43,49,54)(38,45,50,42)(40,46,52,47)(51,55,56,53);(1,2)(3,6)(4,5)(7,8)(9,11,14,20)(10,15,17,18)(12,19,13,21)(16,23,22,24)(25,27)(26,31)(28,37)(29,38)(30,36)(32,45)(33,43)(34,44)(35,42)(39,49)(40,51)(41,50)(46,55)(47,53)(48,54)(52,56);(2,5)(6,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(26,32)(27,36)(29,41)(31,42)(33,46)(34,35)(37,49)(40,52)(43,53)(44,45)(47,48)(54,55);(2,5)(6,8)(10,17)(11,20)(16,22)(19,21)(25,28)(26,33)(27,37)(29,40)(30,39)(31,43)(32,46)(34,48)(35,47)(36,49)(38,51)(41,52)(42,53)(44,54)(45,55)(50,56);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(26,34)(28,39)(31,44)(32,35)(37,49)(40,52)(42,45)(51,56)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfhi,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6561541);