Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667980 |
| 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 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26)(27,32)(28,39)(29,34)(30,44)(31,35)(33,41)(36,53)(37,47)(38,48)(40,56)(42,51)(43,49)(45,52)(46,55)(50,54);(3,4)(6,7)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27)(26,32)(28,40)(29,43)(30,37)(31,38)(33,50)(34,49)(35,48)(36,42)(39,56)(41,54)(44,47)(45,55)(46,52)(51,53);(1,3,5,4)(2,6,8,7)(25,28)(26,33)(27,36)(29,41)(30,45)(31,42)(32,46)(34,51)(35,52)(37,54)(38,40)(39,44)(43,55)(47,53)(48,50)(49,56);(10,15)(14,17)(16,23)(22,24)(25,29,31,30)(26,34,35,44)(27,37,38,43)(28,41,42,45)(32,47,48,49)(33,51,52,39)(36,54,40,55)(46,53,50,56);(1,4,5,3)(2,7,8,6)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(27,38)(32,48)(36,40)(37,43)(46,50)(47,49)(53,56)(54,55)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=hi,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6755394);