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 | |
1318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(12,16)(14,15)(17,18,21,24)(19,25,26,22)(20,23,28,30)(27,31,32,29)(33,34,38,39)(35,40,42,36)(37,41,46,47)(43,48,45,44);(2,5)(6,8)(9,10,13,11)(12,14,16,15)(19,26)(22,25)(27,32)(29,31)(34,39)(35,42)(41,47)(43,45);(1,2,4,5)(3,6,7,8)(9,11,13,10)(12,15,16,14)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)(33,35)(34,40)(36,39)(37,45)(38,42)(41,44)(43,46)(47,48);(9,12,13,16)(10,14,11,15)(17,19,21,26)(18,22,24,25)(20,27,28,32)(23,29,30,31)(33,36,38,40)(34,35,39,42)(37,44,46,48)(41,43,47,45);(1,3)(2,6)(4,7)(5,8)(17,20,21,28)(18,23,24,30)(19,27,26,32)(22,29,25,31)(33,37,38,46)(34,41,39,47)(35,43,42,45)(36,44,40,48)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,c^2=fghi,d^2=fi,e^2=fg,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, 4623577);