Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432843 |
| 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 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1328 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)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,31,35)(27,32,38,47)(28,40,42,48)(29,34,30,43)(33,44,49,41)(36,50,52,55)(37,46,39,53)(45,54,56,51);(2,5)(6,8)(9,11,13,18)(10,14,16,22)(12,19,20,17)(15,23,24,21)(25,27)(26,32)(28,36)(29,37)(30,39)(31,38)(33,45)(34,46)(35,47)(40,50)(41,51)(42,52)(43,53)(44,54)(48,55)(49,56);(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28)(26,33)(27,36)(29,41)(30,44)(31,42)(32,45)(34,48)(35,49)(37,51)(38,52)(39,54)(40,43)(46,55)(47,56)(50,53);(10,16)(14,22)(15,24)(21,23)(25,29,31,30)(26,34,35,43)(27,37,38,39)(28,41,42,44)(32,46,47,53)(33,48,49,40)(36,51,52,54)(45,55,56,50);(1,3,4,7)(2,6,5,8)(27,38)(32,47)(36,52)(37,39)(45,56)(46,53)(50,55)(51,54)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,e^b=ei,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342118);