Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,4,7)(11,15)(13,16)(18,22)(19,25)(20,28)(30,32)(33,34)(35,41)(36,38)(37,44)(39,42)(40,45)(43,47)(46,48);(1,3,5,4)(2,6,8,7)(9,10,12,14)(11,13,15,16)(18,22)(20,28)(23,26)(27,31)(34,36)(35,42)(37,45)(47,48);(1,3,5,4)(2,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,35)(34,39)(36,41)(37,46)(38,42)(40,48)(43,45)(44,47);(1,2)(3,6)(4,7)(5,8)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,36,38,34)(35,41,42,39)(37,44,45,40)(43,48,46,47);(1,4,5,3)(2,7,8,6)(9,11,12,15)(10,13,14,16)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,47)(41,48)(42,46)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=fi,d^2=f,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5113799);