Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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:
(2,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18)(19,25)(20,28)(21,22)(23,27)(24,30)(26,31)(29,32)(33,34,38,41)(35,36,42,39)(37,45,46,40)(43,47,48,44);(9,11,14,12)(10,15,13,16)(18,22)(23,25)(24,28)(31,32)(35,42)(36,39)(43,48)(44,47);(9,12,14,11)(10,15,13,16)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,30)(28,31)(33,35)(34,39)(36,41)(37,43)(38,42)(40,47)(44,45)(46,48);(1,2)(3,6)(4,5)(7,8)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(33,36,38,39)(34,35,41,42)(37,44,46,47)(40,43,45,48);(1,3,4,7)(2,6,5,8)(9,13,14,10)(11,16,12,15)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29)(33,37,38,46)(34,40,41,45)(35,43,42,48)(36,44,39,47)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=fgh,d^2=h,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4669874);