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 | |
1403 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)(28,40)(29,34)(30,44)(31,35)(33,43)(36,49)(37,46)(38,47)(39,53)(41,54)(42,48)(45,51)(50,55)(52,56);(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)(28,36)(29,38)(30,39)(31,37)(33,45)(34,47)(35,46)(40,49)(41,52)(42,50)(43,51)(44,53)(48,55)(54,56);(3,8)(5,7)(12,21)(15,20)(18,24)(22,23)(25,28,30,43,31,41,29,42)(26,33,34,40,35,48,44,54)(27,36,39,51,37,52,38,50)(32,45,47,49,46,55,53,56);(2,6)(5,7)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(27,37)(28,41)(32,46)(33,48)(38,39)(40,54)(42,43)(47,53);(11,19)(14,17)(18,24)(22,23)(25,29,31,30)(26,34,35,44)(27,38,37,39)(28,42,41,43)(32,47,46,53)(33,40,48,54)(36,50,52,51)(45,49,55,56)Polycyclic presentation:
2^9|a^2=fh,c^2=g,d^2=fh,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, 6855659);