Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1265 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,31)(28,33)(29,42)(30,34)(32,41)(35,44)(36,52)(37,46)(38,48)(39,56)(40,51)(43,50)(45,53)(47,55)(49,54);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,30,38)(26,31,34,48)(28,37,40,36)(29,39,44,55)(32,49,50,45)(33,46,51,52)(35,47,42,56)(41,54,43,53);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,30,40)(26,32,34,50)(27,36,38,37)(29,43,44,41)(31,45,48,49)(33,35,51,42)(39,54,55,53)(46,56,52,47);(3,8)(6,7)(12,20)(15,24)(17,19)(21,23)(25,28)(26,33)(27,37)(29,41)(30,40)(31,46)(32,42)(34,51)(35,50)(36,38)(39,54)(43,44)(45,47)(48,52)(49,56)(53,55);(1,4,5,2)(3,7,8,6)(10,16)(11,18)(15,24)(17,19)(26,34)(27,38)(28,40)(35,42)(39,55)(41,43)(45,49)(46,52)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6944507);