Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324820 |
| 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 | |
1372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,24)(21,23)(25,26,30,35)(27,36,37,47)(28,32,41,48)(29,33,38,34)(31,44,45,40)(39,51,52,56)(42,49,43,50)(46,53,55,54);(1,2)(3,7)(4,6)(5,8)(9,11,14,20)(10,15,17,18)(12,19,13,21)(16,23,22,24)(27,37)(28,41)(31,45)(32,48)(36,47)(40,44)(42,43)(49,50);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,27,38,44,30,37,29,40)(26,31,33,47,35,45,34,36)(28,39,42,53,41,52,43,54)(32,46,50,56,48,55,49,51);(2,6)(3,8)(11,20)(15,18)(19,21)(23,24)(25,28)(26,32)(27,39)(29,43)(30,41)(31,46)(33,50)(34,49)(35,48)(36,51)(37,52)(38,42)(40,54)(44,53)(45,55)(47,56);(2,6)(5,7)(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(25,29,30,38)(26,33,35,34)(27,40,37,44)(28,42,41,43)(31,47,45,36)(32,49,48,50)(39,53,52,54)(46,51,55,56)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fgi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dhi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7242309);