Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500015 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
2989 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,9)(6,10)(11,12)(14,15)(17,18)(19,27,30,23)(20,24)(21,25)(22,26)(28,36,31,38)(29,43,46,37)(32,39)(33,40)(34,41)(35,42)(44,47,45,48);(1,2)(3,10)(4,7)(5,8)(6,9)(11,15)(12,14)(13,16)(17,19)(18,23)(20,31)(21,29)(22,30)(24,38)(25,37)(26,27)(28,33)(32,45)(34,44)(35,46)(36,40)(39,48)(41,47)(42,43);(1,3,5,9)(2,6,8,10)(4,12,13,11)(7,15,16,14)(17,20,22,33)(18,24,26,40)(19,28,30,31)(21,34,35,32)(23,36,27,38)(25,41,42,39)(29,45,46,44)(37,48,43,47);(1,4,5,13)(2,7,8,16)(3,11,9,12)(6,14,10,15)(17,21,22,35)(18,25,26,42)(19,29,30,46)(20,32,33,34)(23,37,27,43)(24,39,40,41)(28,44,31,45)(36,47,38,48);(2,8)(6,10)(7,16)(14,15)(19,30)(21,35)(23,27)(25,42)(28,31)(32,34)(36,38)(39,41)Polycyclic presentation:
2^9|a^2=ghi,c^2=h,d^2=h,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, 6439805);