Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682596 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1322 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)(3,8)(4,6)(5,7)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,30,34)(27,31,36,44)(28,38,39,53)(29,41,42,33)(32,46,47,40)(35,48,49,56)(37,52,51,45)(43,54,55,50);(3,4)(6,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27)(26,31)(28,35)(29,37)(30,36)(32,43)(33,45)(34,44)(38,48)(39,49)(40,50)(41,52)(42,51)(46,54)(47,55)(53,56);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,29,40,30,39,42,46)(26,32,33,38,34,47,41,53)(27,35,37,50,36,49,51,54)(31,43,45,48,44,55,52,56);(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(27,36)(28,39)(31,44)(32,47)(37,51)(38,53)(40,46)(45,52);(1,4,5,3)(2,6,7,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(28,39)(32,47)(35,49)(38,53)(40,46)(43,55)(48,56)(50,54)Polycyclic presentation:
2^9|a^2=i,c^2=fg,d^2=f,e^2=fh,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=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6705674);