Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884969 |
| 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 | |
1401 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)(11,16)(12,21)(13,17)(14,18)(15,20)(19,23)(22,24)(25,26,31,35)(27,32,30,43)(28,38,40,45)(29,34,42,50)(33,44,48,36)(37,46,41,49)(39,52,54,55)(47,53,56,51);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,19)(13,22)(14,20)(16,18)(17,23)(21,24)(25,27,31,30)(26,32,35,43)(28,36,40,44)(29,41,42,37)(33,45,48,38)(34,49,50,46)(39,53,54,51)(47,52,56,55);(3,8)(5,7)(9,11)(10,16)(12,22)(13,19)(14,20)(15,18)(17,23)(21,24)(25,28,31,40)(26,33,35,48)(27,36,30,44)(29,39,42,54)(32,45,43,38)(34,47,50,56)(37,51,41,53)(46,55,49,52);(2,6)(5,7)(9,12,14,13)(10,17,18,21)(11,19,20,22)(15,23,16,24)(25,29)(26,34)(27,37)(28,39)(30,41)(31,42)(32,46)(33,47)(35,50)(36,51)(38,52)(40,54)(43,49)(44,53)(45,55)(48,56);(9,13,14,12)(10,17,18,21)(11,19,20,22)(15,24,16,23)(26,35)(29,42)(32,43)(33,48)(37,41)(38,45)(39,54)(51,53)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6893491);