Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907909 |
| 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 | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1405 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26)(27,32)(28,39)(29,34)(30,45)(31,35)(33,42)(36,52)(37,47)(38,48)(40,54)(41,51)(43,55)(44,49)(46,53)(50,56);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27)(26,32)(28,40)(29,44)(30,38)(31,37)(33,50)(34,49)(35,47)(36,43)(39,54)(41,53)(42,56)(45,48)(46,51)(52,55);(3,8)(5,7)(12,21)(15,20)(18,24)(22,23)(25,28,29,41,31,43,30,42)(26,33,45,55,35,51,34,39)(27,36,44,56,37,40,38,53)(32,46,48,54,47,50,49,52);(2,6)(5,7)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(27,37)(32,47)(36,40)(38,44)(46,50)(48,49)(52,54)(53,56);(11,19)(14,17)(18,24)(22,23)(25,29,31,30)(26,34,35,45)(27,38,37,44)(28,41,43,42)(32,48,47,49)(33,39,51,55)(36,53,40,56)(46,54,50,52)Polycyclic presentation:
2^9|a^2=fhi,c^2=gi,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfhi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6855668);