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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,3)(4,8)(11,15)(13,16)(17,18)(19,22)(20,25)(21,26)(23,27)(24,29)(28,30)(31,32)(33,34,38,39)(35,40,41,36)(37,45,46,42)(43,44,48,47);(2,3)(6,7)(9,10,12,14)(11,13,15,16)(18,19)(20,27)(21,29)(30,31)(34,39)(35,41)(37,46)(44,47);(1,2,5,3)(4,6,8,7)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,35)(34,40)(36,39)(37,43)(38,41)(42,47)(44,45)(46,48);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,15)(14,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,36,38,40)(34,41,39,35)(37,44,46,47)(42,48,45,43);(1,4,5,8)(2,6,3,7)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,46)(34,42,39,45)(35,43,41,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=g,d^2=h,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, 4384588);