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 | |
1260 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)(25,26,29,34)(27,36,38,31)(28,33,41,52)(30,44,43,35)(32,45,50,42)(37,51,40,47)(39,48,55,53)(46,54,49,56);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,29,38)(26,31,34,36)(28,40,41,37)(30,39,43,55)(32,49,50,46)(33,51,52,47)(35,48,44,53)(42,56,45,54);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,29,41)(26,32,34,50)(27,37,38,40)(30,45,43,42)(31,46,36,49)(33,35,52,44)(39,56,55,54)(47,48,51,53);(3,8)(6,7)(12,20)(15,24)(17,19)(21,23)(25,28)(26,33)(27,37)(29,41)(30,42)(31,47)(32,44)(34,52)(35,50)(36,51)(38,40)(39,54)(43,45)(46,53)(48,49)(55,56);(1,4,5,2)(3,7,8,6)(10,16)(11,18)(15,24)(17,19)(26,34)(28,41)(31,36)(35,44)(37,40)(42,45)(48,53)(54,56)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,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, 6944636);