Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352469 |
| 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 |
| prime radical | #62867 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2743 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)(9,13)(10,12)(14,16)(17,18,23,28)(19,29,32,42)(20,25,33,39)(21,27,22,26)(24,36,38,31)(30,43,44,48)(34,41,35,40)(37,46,47,45);(1,2)(3,8)(4,7)(5,13)(6,9)(10,14)(11,16)(12,15)(17,19)(18,24)(20,30)(21,31)(22,36)(23,32)(25,37)(26,29)(27,42)(28,38)(33,44)(34,45)(35,46)(39,47)(40,43)(41,48);(1,3,6,11)(2,8,9,16)(4,10,5,12)(7,14,13,15)(17,20,23,33)(18,25,28,39)(19,30,32,44)(21,34,22,35)(24,37,38,47)(26,40,27,41)(29,43,42,48)(31,45,36,46);(1,3)(2,8)(4,12)(5,10)(6,11)(7,15)(9,16)(13,14)(20,33)(25,39)(30,44)(34,35)(37,47)(40,41)(43,48)(45,46);(1,4,6,5)(2,7,9,13)(3,10,11,12)(8,14,16,15)(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,34,33,35)(24,29,38,42)(25,40,39,41)(30,45,44,46)(37,43,47,48)Polycyclic presentation:
2^9|a^2=hi,c^2=h,e^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259758);