Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10,13,11)(12,15,16,14)(19,25)(20,27)(23,28)(24,29)(33,34,38,40)(35,41,43,36)(37,39,45,46)(42,47,48,44);(1,2)(3,8)(4,5)(6,7)(9,10,13,11)(12,14,16,15)(18,22)(23,28)(24,29)(30,32)(37,45)(39,46)(42,48)(44,47);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,38,43)(34,36,40,41)(37,42,45,48)(39,44,46,47);(1,2)(3,8)(4,5)(6,7)(9,12)(10,14)(11,15)(13,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48);(1,3)(2,6)(4,7)(5,8)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,37)(34,39)(35,42)(36,44)(38,45)(40,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=fhi,c^2=gi,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5699141);