Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,8)(4,6)(5,7)(9,10,13,16)(11,17,18,14)(12,15,20,22)(19,23,24,21)(25,26)(27,36)(28,40)(29,34)(30,35)(31,44)(32,39)(33,41)(37,52)(38,48)(42,49)(43,47)(45,50)(46,53)(51,55)(54,56);(1,3,5,4)(2,6,7,8)(9,11,13,18)(10,14,16,17)(12,19,20,24)(15,21,22,23)(25,27)(26,32)(28,37)(29,43)(30,39)(31,38)(33,46)(34,48)(35,36)(40,56)(41,55)(42,54)(44,47)(45,53)(49,52)(50,51);(11,18)(14,17)(19,24)(21,23)(25,28,30,42)(26,33,35,50)(27,37,39,54)(29,41,31,45)(32,46,36,51)(34,49,44,40)(38,53,43,55)(47,56,48,52);(1,4,5,3)(2,6,7,8)(12,20)(15,22)(19,24)(21,23);(9,12,13,20)(10,15,16,22)(11,19,18,24)(14,21,17,23)(25,29,30,31)(26,34,35,44)(27,38,39,43)(28,41,42,45)(32,47,36,48)(33,49,50,40)(37,53,54,55)(46,56,51,52)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=i,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ei,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6473227);