Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470149 |
| 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 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1379 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26)(27,36)(28,40)(29,33)(30,35)(31,39)(32,42)(34,45)(37,46)(38,48)(41,49)(43,50)(44,47)(51,54)(52,55)(53,56);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,30,38)(26,31,35,47)(28,37,43,52)(29,44,45,39)(32,46,49,55)(33,36,34,48)(40,51,50,56)(41,54,42,53);(1,4,5,3)(2,6,7,8)(11,17)(14,21)(18,19)(22,23)(25,28)(26,32)(27,37)(29,41)(30,43)(31,46)(33,40)(34,50)(35,49)(36,51)(38,52)(39,53)(42,45)(44,54)(47,55)(48,56);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(27,38)(31,47)(36,48)(37,52)(39,44)(46,55)(51,56)(53,54);(10,16)(14,21)(15,24)(22,23)(25,29,30,45)(26,33,35,34)(27,39,38,44)(28,41,43,42)(31,48,47,36)(32,40,49,50)(37,53,52,54)(46,56,55,51)Polycyclic presentation:
2^9|b^2=fg,c^2=fgi,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6392231);