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 | |
1298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,5)(3,8,4,7)(9,10,13,11)(12,15,16,14)(19,25)(20,27)(23,28)(24,29)(33,34)(35,39)(36,44)(37,38)(40,45)(41,42)(43,48)(46,47);(1,2)(3,7)(4,8)(5,6)(9,11,13,10)(12,15,16,14)(18,22)(23,28)(24,29)(30,32)(34,38)(35,41)(40,44)(43,47);(1,3,6,4)(2,7,5,8)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,34)(35,42)(36,44)(37,38)(39,41)(40,45)(43,46)(47,48);(1,4,6,3)(2,7,5,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,48)(45,47);(1,5,6,2)(3,7,4,8)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,36,37,45)(34,40,38,44)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|a^2=fhi,b^2=fghi,c^2=g,d^2=g,e^2=gh,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, 4982990);