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 | #3459 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
2831 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(17,18,23,28)(19,24,32,39)(20,25,35,42)(21,27,22,26)(29,43,46,36)(30,38,31,37)(33,41,34,40)(44,47,45,48);(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,19)(18,24)(20,29)(21,30)(22,31)(23,32)(25,36)(26,37)(27,38)(28,39)(33,44)(34,45)(35,46)(40,47)(41,48)(42,43);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,20)(18,25)(19,29)(21,33)(22,34)(23,35)(24,36)(26,40)(27,41)(28,42)(30,44)(31,45)(32,46)(37,47)(38,48)(39,43);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,21,23,22)(18,26,28,27)(19,30,32,31)(20,33,35,34)(24,37,39,38)(25,40,42,41)(29,44,46,45)(36,47,43,48);(4,12)(7,14)(10,16)(13,15)(17,22,23,21)(18,27,28,26)(19,31,32,30)(20,34,35,33)(24,38,39,37)(25,41,42,40)(29,45,46,44)(36,48,43,47)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=i,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,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, 6478818);