Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500015 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | #62866 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
2970 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,8)(5,7)(9,13)(10,12)(11,14)(15,16)(17,18)(19,27,30,23)(20,31)(21,25)(22,26)(24,33)(28,39,42,35)(29,40,43,36)(32,44)(34,38)(37,45)(41,47,48,46);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,35)(25,36)(26,27)(31,39)(32,41)(33,42)(34,43)(37,46)(38,40)(44,47)(45,48);(2,7)(3,9)(4,11)(12,16)(17,20)(18,24)(19,28)(21,32)(22,33)(23,35)(25,37)(26,31)(27,39)(29,41)(30,42)(34,45)(36,46)(38,44)(40,47)(43,48);(1,4,5,11)(2,8,7,14)(3,10,9,15)(6,12,13,16)(21,34)(25,38)(29,43)(32,45)(36,40)(37,44)(41,48)(46,47);(3,9)(6,13)(10,15)(12,16)(17,21,22,34)(18,25,26,38)(19,29,30,43)(20,32,33,45)(23,36,27,40)(24,37,31,44)(28,41,42,48)(35,46,39,47)Polycyclic presentation:
2^9|a^2=ghi,d^2=h,e^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6432010);