Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200042 |
| 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×#127 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1373 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)(4,5)(7,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,22)(19,23)(21,24)(25,26,30,34)(27,31,36,45)(28,39,40,33)(29,32,42,35)(37,51,49,44)(38,46,50,43)(41,48,53,47)(52,54,56,55);(2,5)(3,7)(9,11,13,18)(10,14,15,17)(12,21,22,19)(16,24,20,23)(25,27,30,36)(26,31,34,45)(28,37,40,49)(29,38,42,50)(32,46,35,43)(33,44,39,51)(41,52,53,56)(47,55,48,54);(11,18)(14,17)(19,21)(23,24)(26,32)(27,36)(29,42)(31,43)(33,47)(34,35)(37,49)(39,48)(41,53)(44,54)(45,46)(51,55);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,20)(23,24)(25,28,30,40)(26,33,34,39)(27,37,36,49)(29,41,42,53)(31,44,45,51)(32,47,35,48)(38,52,50,56)(43,54,46,55);(2,5)(6,8)(9,12,13,22)(10,16,15,20)(11,19,18,21)(14,23,17,24)(26,34)(27,36)(28,40)(32,35)(38,50)(41,53)(44,51)(54,55)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=efi,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7140542);