Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1233 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,6,8,7)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26)(27,35)(28,39)(29,31)(30,44)(32,38)(33,41)(34,43)(36,50)(37,52)(40,49)(42,48)(45,51)(46,56)(47,54)(53,55);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(26,31)(27,36)(33,39)(34,44)(37,40)(38,51)(48,55)(54,56);(1,2,5,4)(3,7,8,6)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,27,30,38,29,36,43,51)(26,32,34,35,31,45,44,50)(28,40,42,56,41,37,53,54)(33,47,48,52,39,46,55,49);(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,28)(26,33)(27,37)(29,41)(30,42)(31,39)(32,46)(34,48)(35,49)(36,40)(38,54)(43,53)(44,55)(45,47)(50,52)(51,56);(1,4,5,2)(3,6,8,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(28,41)(33,39)(37,40)(42,53)(46,47)(48,55)(49,52)(54,56)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=fgh,d^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6765699);