Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433057 |
| 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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1239 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)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,32)(28,38)(29,42)(30,45)(31,35)(33,39)(34,44)(36,49)(37,50)(40,54)(41,47)(43,46)(48,51)(52,55)(53,56);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,31,30)(26,32,35,45)(28,39,41,36)(29,43,44,37)(33,47,49,38)(34,50,42,46)(40,51,55,53)(48,52,56,54);(11,17)(14,21)(18,23)(22,24)(25,28,30,39,31,41,27,36)(26,33,32,38,35,49,45,47)(29,40,43,53,44,55,37,51)(34,48,46,52,42,56,50,54);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,29,31,44)(26,34,35,42)(27,37,30,43)(28,40,41,55)(32,46,45,50)(33,48,49,56)(36,51,39,53)(38,52,47,54);(1,3,4,7)(2,6,5,8)(29,44)(34,42)(37,43)(40,55)(46,50)(48,56)(51,53)(52,54)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=g,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^b=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343630);