Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,7)(3,8,5,4)(11,15)(13,16)(17,18,21,24)(19,25,26,22)(20,23,28,30)(27,31,32,29)(33,34,39,42)(35,37,36,40)(38,41,46,48)(43,45,44,47);(2,7)(3,5)(19,26)(22,25)(27,32)(29,31)(33,35,39,36)(34,40,42,37)(38,44,46,43)(41,45,48,47);(1,3,6,5)(2,8,7,4)(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)(33,36,39,35)(34,37,42,40)(38,44,46,43)(41,45,48,47);(1,4,6,8)(2,3,7,5)(9,11,12,15)(10,13,14,16)(17,19)(18,22)(20,27)(21,26)(23,29)(24,25)(28,32)(30,31)(33,37)(34,36)(35,42)(38,45)(39,40)(41,44)(43,48)(46,47);(1,5,6,3)(2,8,7,4)(17,20,21,28)(18,23,24,30)(19,27,26,32)(22,29,25,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=gh,b^2=fh,c^2=ghi,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4430870);