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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1301 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)(25,26,30,34)(27,35,36,48)(28,32,40,46)(29,42,37,33)(31,43,44,39)(38,49,50,56)(41,53,51,47)(45,54,55,52);(1,2)(3,8)(4,6)(5,7)(9,11,13,18)(10,14,16,22)(12,19,20,17)(15,23,24,21)(27,36)(31,44)(35,48)(38,50)(39,43)(45,55)(49,56)(52,54);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,37,43,30,36,29,39)(26,31,42,48,34,44,33,35)(28,38,51,54,40,50,41,52)(32,45,53,56,46,55,47,49);(1,3,5,4)(2,6,7,8)(12,20)(15,24)(17,19)(21,23)(25,28)(26,32)(27,38)(29,41)(30,40)(31,45)(33,47)(34,46)(35,49)(36,50)(37,51)(39,52)(42,53)(43,54)(44,55)(48,56);(1,4,5,3)(2,6,7,8)(11,18)(14,22)(17,19)(21,23)(27,36)(28,40)(31,44)(32,46)(35,48)(39,43)(41,51)(47,53)Polycyclic presentation:
2^9|a^2=fh,b^2=hi,c^2=gi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,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, 7178636);