Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234485 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1320 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)(3,6)(4,7)(5,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,35)(28,33)(29,31)(30,42)(32,39)(34,36)(37,48)(38,47)(40,44)(41,53)(43,49)(45,52)(46,50)(51,55)(54,56);(1,3,6,2)(4,7,5,8)(9,11,13,18)(10,14,16,22)(12,19,20,17)(15,23,24,21)(26,31)(28,40)(32,43)(34,42)(35,47)(37,51)(41,50)(52,56);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,36,49,29,38,30,39)(26,32,42,47,31,43,34,35)(28,37,50,56,40,51,41,52)(33,45,53,55,44,54,46,48);(1,4,6,5)(2,7,3,8)(12,20)(15,24)(17,19)(21,23)(25,28,29,40)(26,33,31,44)(27,37,38,51)(30,41,36,50)(32,45,43,54)(34,46,42,53)(35,48,47,55)(39,52,49,56);(1,5,6,4)(2,7,3,8)(11,18)(14,22)(17,19)(21,23)(26,31)(27,38)(28,40)(34,42)(39,49)(41,50)(45,54)(48,55)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=gi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7178749);