Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351817 |
| 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 | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,12)(5,7)(6,11)(8,13)(10,16)(14,15)(17,18)(19,29)(20,33)(21,27)(22,28)(23,26)(24,31)(25,35)(30,38)(32,40)(34,41)(36,39)(37,42)(43,46)(44,48)(45,47);(4,13)(8,12)(9,16)(11,14)(17,19)(18,24)(20,30)(21,36)(22,31)(23,32)(25,38)(26,39)(27,29)(28,40)(33,47)(34,44)(35,46)(37,45)(41,43)(42,48);(1,3)(2,6)(4,14)(5,10)(7,15)(8,16)(9,12)(11,13)(17,20)(18,25)(19,30)(21,35)(22,37)(23,34)(24,38)(26,42)(27,41)(28,33)(29,43)(31,45)(32,44)(36,46)(39,48)(40,47);(2,7)(6,15)(8,12)(9,16)(18,26)(20,34)(24,39)(27,28)(29,40)(30,44)(35,37)(45,46);(17,21,23,22)(18,27,26,28)(19,31,32,36)(20,35,34,37)(24,40,39,29)(25,41,42,33)(30,45,44,46)(38,47,48,43)Polycyclic presentation:
2^9|e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,g^a=ghi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=ghi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259019);