Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500443 |
| 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 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,7)(3,8,4,6)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,32)(28,39)(29,34)(30,44)(31,35)(33,42)(36,50)(37,46)(38,47)(40,49)(41,53)(43,48)(45,51)(52,55)(54,56);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27)(26,32)(28,36)(29,43)(30,37)(31,38)(33,45)(34,48)(35,47)(39,50)(40,54)(41,52)(42,51)(44,46)(49,56)(53,55);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,30,42,31,41,29,40)(26,33,34,39,35,49,44,53)(27,36,37,51,38,52,43,54)(32,45,48,50,47,56,46,55);(12,20)(15,24)(17,19)(21,23)(25,29,31,30)(26,34,35,44)(27,37,38,43)(28,40,41,42)(32,46,47,48)(33,39,49,53)(36,51,52,54)(45,55,56,50);(1,4,5,3)(2,6,7,8)(11,18)(14,22)(17,19)(21,23)(27,38)(28,41)(32,47)(33,49)(37,43)(39,53)(40,42)(46,48)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=g,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6419770);