Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| 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 | |
1362 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)(25,26,29,34)(27,36,38,31)(28,40,41,33)(30,32,43,35)(37,45,50,49)(39,46,52,44)(42,48,53,47)(51,54,56,55);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27)(26,31)(28,37)(29,38)(30,39)(32,44)(33,45)(34,36)(35,46)(40,49)(41,50)(42,51)(43,52)(47,54)(48,55)(53,56);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(26,32)(28,41)(30,43)(31,44)(33,48)(34,35)(36,46)(37,50)(39,52)(40,47)(45,55)(49,54);(3,8)(5,7)(10,15)(14,17)(16,23)(22,24)(25,28)(26,33)(27,37)(29,41)(30,42)(31,45)(32,47)(34,40)(35,48)(36,49)(38,50)(39,51)(43,53)(44,54)(46,55)(52,56);(2,6)(5,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(27,38)(28,41)(31,36)(33,40)(39,52)(42,53)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=i,b^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,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, 6796926);