Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(9,10,13,11)(12,15,16,14)(17,18,21,22)(19,25,26,23)(20,24,28,30)(27,32,29,31)(33,34)(35,42)(36,40)(37,39)(38,45)(41,46)(43,47)(44,48);(2,5)(6,8)(18,22)(19,26)(20,28)(31,32)(33,35,39,36)(34,40,37,42)(38,44,46,43)(41,48,45,47);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,18)(19,25)(20,24)(21,22)(23,26)(27,32)(28,30)(29,31)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,14,11,15)(17,19)(18,23)(20,29)(21,26)(22,25)(24,32)(27,28)(30,31)(33,37,39,34)(35,42,36,40)(38,41,46,45)(43,47,44,48);(17,20)(18,24)(19,27)(21,28)(22,30)(23,31)(25,32)(26,29)(33,38,39,46)(34,41,37,45)(35,43,36,44)(40,47,42,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=fg,c^2=gi,d^2=hi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3097052);