Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533141 |
| 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 | #62903 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
2861 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(17,18)(19,27)(20,31,33,24)(21,26)(22,25)(23,30)(28,39,42,35)(29,40)(32,44,45,37)(34,38)(36,43)(41,47,48,46);(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,19)(18,23)(20,28)(21,29)(22,30)(24,35)(25,27)(26,36)(31,39)(32,41)(33,42)(34,43)(37,46)(38,40)(44,47)(45,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,20)(18,24)(19,28)(21,32)(22,33)(23,35)(25,31)(26,37)(27,39)(29,41)(30,42)(34,45)(36,46)(38,44)(40,47)(43,48);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(18,25)(23,27)(24,31)(26,38)(35,39)(36,40)(37,44)(46,47);(4,12)(7,14)(10,16)(13,15)(17,21,22,34)(18,26,25,38)(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=fi,b^2=h,c^2=h,e^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,g^b=gi,f^c=fi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6479828);