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 | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1215 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,30,35)(27,32,40,49)(28,41,44,33)(29,34,31,36)(37,53,55,46)(38,47,39,48)(42,56,54,50)(43,52,45,51);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27)(26,32)(28,42)(29,38)(30,40)(31,39)(33,50)(34,47)(35,49)(36,48)(37,43)(41,56)(44,54)(45,55)(46,51)(52,53);(2,5)(6,8)(25,28,30,44)(26,33,35,41)(27,37,40,55)(29,43,31,45)(32,46,49,53)(34,51,36,52)(38,54,39,42)(47,56,48,50);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,23)(22,24)(25,29,30,31)(26,34,35,36)(27,38,40,39)(28,43,44,45)(32,47,49,48)(33,51,41,52)(37,54,55,42)(46,56,53,50);(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(27,39)(29,31)(32,48)(34,36)(37,42)(38,40)(43,45)(46,50)(47,49)(51,52)(53,56)(54,55)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6611220);