Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(11,15)(14,16)(17,18,22,20)(19,25,26,23)(21,28,29,24)(27,30,32,31)(33,34,38,40)(35,41,43,36)(37,39,46,48)(42,47,45,44);(10,13)(14,16)(19,26)(21,29)(23,25)(24,28)(33,35,38,43)(34,36,40,41)(37,45,46,42)(39,47,48,44);(9,10)(11,14)(12,13)(15,16)(17,19,22,26)(18,23,20,25)(21,27,29,32)(24,30,28,31)(33,35)(34,36)(37,45)(38,43)(39,47)(40,41)(42,46)(44,48);(1,2)(3,6)(4,5)(7,8)(17,20,22,18)(19,23,26,25)(21,28,29,24)(27,30,32,31)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,48)(46,47);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37)(34,39)(35,42)(36,44)(38,46)(40,48)(41,47)(43,45)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fg,d^2=fg,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,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, 5267209);