Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470231 |
| 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 | #2404 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
2871 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)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,18,22,26)(19,23,28,36)(20,24,32,40)(21,33,34,25)(27,38,30,35)(29,43,42,37)(31,45,46,39)(41,47,44,48);(1,3,6,13)(2,7,10,16)(4,12,5,11)(8,15,9,14)(17,19,22,28)(18,23,26,36)(20,30,32,27)(21,29,34,42)(24,38,40,35)(25,37,33,43)(31,44,46,41)(39,48,45,47);(1,4,6,5)(2,8,10,9)(3,11,13,12)(7,14,16,15)(17,20,22,32)(18,24,26,40)(19,27,28,30)(21,31,34,46)(23,35,36,38)(25,39,33,45)(29,41,42,44)(37,47,43,48);(1,5,6,4)(2,9,10,8)(3,12,13,11)(7,15,16,14)(19,28)(21,34)(23,36)(25,33)(27,30)(31,46)(35,38)(39,45);(2,10)(7,16)(8,9)(14,15)(17,21,22,34)(18,25,26,33)(19,29,28,42)(20,31,32,46)(23,37,36,43)(24,39,40,45)(27,41,30,44)(35,47,38,48)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=hi,e^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,g^b=gi,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, 6396960);