Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6353061 |
| 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 | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2773 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,7)(4,5)(8,15)(9,12)(10,13)(14,16)(17,18,23,27)(19,28,31,38)(20,25,22,36)(21,33,34,26)(24,37,40,29)(30,44,45,42)(32,46,35,41)(39,43,47,48);(1,2)(3,8)(4,7)(5,12)(6,9)(10,14)(11,16)(13,15)(17,19)(18,24)(20,29)(21,30)(22,37)(23,31)(25,38)(26,39)(27,40)(28,36)(32,43)(33,47)(34,45)(35,48)(41,44)(42,46);(1,3,6,11)(2,8,9,16)(4,10,5,13)(7,14,12,15)(17,20,23,22)(18,25,27,36)(19,29,31,37)(21,32,34,35)(24,38,40,28)(26,41,33,46)(30,43,45,48)(39,44,47,42);(1,4,6,5)(2,7,9,12)(3,10,11,13)(8,14,16,15)(17,21,23,34)(18,26,27,33)(19,30,31,45)(20,32,22,35)(24,39,40,47)(25,41,36,46)(28,42,38,44)(29,43,37,48);(21,34)(26,33)(30,45)(32,35)(39,47)(41,46)(42,44)(43,48)Polycyclic presentation:
2^9|a^2=hi,c^2=h,d^2=h,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ehi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260458);