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×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1312 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)(3,7)(4,8)(5,6)(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26)(27,36)(28,39)(29,34)(30,35)(31,32)(33,40)(37,49)(38,50)(41,55)(42,46)(43,47)(44,56)(45,52)(48,54)(51,53);(2,6)(3,4)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(25,27,30,31)(26,32,35,36)(28,40,42,37)(29,43,45,38)(33,39,49,46)(34,50,52,47)(41,53,44,54)(48,56,51,55);(1,2)(3,8)(4,7)(5,6)(25,28,27,37,30,42,31,40)(26,33,32,46,35,49,36,39)(29,44,38,54,45,41,43,53)(34,51,47,55,52,48,50,56);(1,3,5,4)(2,7,6,8)(11,18)(14,17)(19,24)(22,23)(25,29,30,45)(26,34,35,52)(27,38,31,43)(28,41,42,44)(32,47,36,50)(33,48,49,51)(37,53,40,54)(39,55,46,56);(1,4,5,3)(2,7,6,8)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(26,35)(28,42)(29,45)(32,36)(37,40)(38,43)(48,51)(55,56)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=gi,d^2=h,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=ehi,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, 7200029);