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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1319 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,7)(3,6,4,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,30,35)(27,36,31,32)(28,39,41,46)(29,34,45,52)(33,37,49,42)(38,50,43,47)(40,55,44,56)(48,53,51,54);(1,2)(3,8)(4,6)(5,7)(9,11,13,18)(10,14,16,22)(12,19,20,17)(15,23,24,21)(25,27,30,31)(26,32,35,36)(28,37,41,42)(29,43,45,38)(33,46,49,39)(34,50,52,47)(40,54,44,53)(48,55,51,56);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,27,37,30,41,31,42)(26,33,32,46,35,49,36,39)(29,44,38,54,45,40,43,53)(34,51,47,55,52,48,50,56);(1,3,5,4)(2,6,7,8)(12,20)(15,24)(17,19)(21,23)(25,29)(26,34)(27,38)(28,40)(30,45)(31,43)(32,47)(33,48)(35,52)(36,50)(37,53)(39,55)(41,44)(42,54)(46,56)(49,51);(1,4,5,3)(2,6,7,8)(11,18)(14,22)(17,19)(21,23)(26,35)(28,41)(29,45)(32,36)(37,42)(38,43)(48,51)(55,56)Polycyclic presentation:
2^9|a^2=fh,b^2=hi,c^2=gi,d^2=fi,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=dhi,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, 7178591);