Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234321 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1191 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(3,7)(5,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,32,36,31,37,30,38)(27,39,47,51,42,33,41,50)(28,43,44,34,29,35,45,52)(40,53,55,54,46,49,56,48);(1,2)(3,8)(4,6)(5,7)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27)(26,33)(28,40)(29,46)(30,41)(31,42)(32,47)(34,48)(35,53)(36,50)(37,39)(38,51)(43,49)(44,55)(45,56)(52,54);(1,3,4,7)(2,5,6,8)(25,28,31,29)(26,34,37,52)(27,40,42,46)(30,44,32,45)(33,48,39,54)(35,36,43,38)(41,55,47,56)(49,51,53,50);(3,7)(5,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,29,31,28)(26,35,37,43)(27,40,42,46)(30,44,32,45)(33,49,39,53)(34,36,52,38)(41,56,47,55)(48,51,54,50);(2,6)(5,8)(10,16)(14,17)(15,24)(22,23)(25,30,31,32)(26,36,37,38)(27,41,42,47)(28,44,29,45)(33,50,39,51)(34,43,52,35)(40,55,46,56)(48,49,54,53)Polycyclic presentation:
2^9|a^2=ghi,b^2=fi,c^2=hi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7180845);