Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1411 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)(3,8,4,6)(9,10)(11,17)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24)(25,26,29,33,30,34,44,50)(27,35,37,46,38,49,43,31)(28,39,41,48,42,47,40,32)(36,45,54,56,53,55,52,51);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(25,27,30,38)(26,31,34,46)(28,36,42,53)(29,43,44,37)(32,45,48,55)(33,49,50,35)(39,51,47,56)(40,54,41,52);(1,4,5,3)(2,6,7,8)(10,15)(11,18)(16,22)(19,24)(25,28)(26,32)(27,36)(29,40)(30,42)(31,45)(33,47)(34,48)(35,51)(37,52)(38,53)(39,50)(41,44)(43,54)(46,55)(49,56);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29,30,44)(26,33,34,50)(27,37,38,43)(28,40,42,41)(31,35,46,49)(32,47,48,39)(36,52,53,54)(45,51,55,56);(12,20)(16,22)(19,24)(21,23)(27,38)(31,46)(35,49)(36,53)(37,43)(45,55)(51,56)(52,54)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=fgi,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6477953);