Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469995 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1326 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)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26,30,32)(27,35,36,50)(28,33,40,47)(29,42,44,34)(31,43,46,38)(37,51,39,53)(41,56,55,49)(45,54,48,52);(9,11,14,21)(10,15,18,24)(12,20,13,19)(16,23,17,22)(25,27,30,36)(26,31,32,46)(28,39,40,37)(29,43,44,38)(33,48,47,45)(34,50,42,35)(41,52,55,54)(49,51,56,53);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,24,23)(26,32)(27,36)(33,47)(34,42)(37,39)(38,43)(49,56)(52,54);(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,23,24,22)(25,28,30,40)(26,33,32,47)(27,37,36,39)(29,41,44,55)(31,45,46,48)(34,49,42,56)(35,51,50,53)(38,52,43,54);(1,3,4,7)(2,6,5,8)(10,18)(15,24)(16,17)(22,23)(26,32)(28,40)(31,46)(34,42)(35,50)(37,39)(41,55)(52,54)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=hi,d^2=h,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6397610);