Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1157 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26,30,34)(27,31,38,47)(28,32,40,49)(29,33,42,51)(35,44,37,46)(36,45,54,56)(39,48,53,55)(41,50,43,52);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27,30,38)(26,31,34,47)(28,39,40,53)(29,35,42,37)(32,48,49,55)(33,44,51,46)(36,43,54,41)(45,52,56,50);(2,5)(6,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(27,35)(28,40)(29,42)(31,44)(32,49)(33,51)(36,53)(37,38)(39,54)(45,55)(46,47)(48,56);(1,3,4,7)(2,6,5,8)(10,16)(14,17)(15,22)(21,23)(28,40)(32,49)(36,54)(39,53)(41,43)(45,56)(48,55)(50,52);(11,19)(14,17)(18,24)(21,23)(25,28,30,40)(26,32,34,49)(27,36,38,54)(29,43,42,41)(31,45,47,56)(33,52,51,50)(35,39,37,53)(44,48,46,55)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=f,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6605657);