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 | |
1296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,3)(4,8)(11,15)(13,16)(17,18)(19,22)(20,25)(21,26)(23,27)(24,29)(28,30)(31,32)(33,34,39,42)(35,37,36,40)(38,41,46,48)(43,45,44,47);(2,3)(6,7)(18,19)(20,27)(21,29)(30,31)(33,35,39,36)(34,40,42,37)(38,44,46,43)(41,45,48,47);(1,2,5,3)(4,6,8,7)(9,10,12,14)(11,13,15,16)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,36,39,35)(34,37,42,40)(38,44,46,43)(41,45,48,47);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,13,14,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,37)(34,36)(35,42)(38,45)(39,40)(41,44)(43,48)(46,47);(1,4,5,8)(2,6,3,7)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(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=gh,c^2=ghi,d^2=i,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, 4430668);