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 | |
1300 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)(17,18,22,24)(19,25,26,20)(21,29,30,23)(27,28,32,31)(33,34,38,36)(35,41,43,39)(37,44,46,40)(42,47,45,48);(2,7)(3,5)(19,26)(20,25)(27,32)(28,31)(33,35,38,43)(34,39,36,41)(37,45,46,42)(40,48,44,47);(1,3,6,5)(2,8,7,4)(9,10,12,14)(11,13,15,16)(17,19)(18,20)(21,27)(22,26)(23,28)(24,25)(29,31)(30,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,48)(45,46);(1,4,6,8)(2,3,7,5)(9,11,12,15)(10,13,14,16)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31)(33,36,38,34)(35,39,43,41)(37,44,46,40)(42,47,45,48);(1,5,6,3)(2,8,7,4)(17,21,22,30)(18,23,24,29)(19,27,26,32)(20,28,25,31)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=ghi,d^2=fi,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, 4430646);