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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1429 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,8)(4,7)(5,6)(10,12)(14,16)(17,18,21,24)(19,25,26,22)(20,23,28,30)(27,31,32,29)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(1,3,6,4)(2,7,5,8)(9,10,13,12)(11,14,15,16)(19,26)(22,25)(27,32)(29,31)(34,35)(39,41)(40,42)(46,47);(1,3,6,4)(2,8,5,7)(9,11)(10,14)(12,16)(13,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)(33,35,38,34)(36,41,44,39)(37,40,45,42)(43,46,48,47);(1,4,6,3)(2,8,5,7)(17,19,21,26)(18,22,24,25)(20,27,28,32)(23,29,30,31)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,5,6,2)(3,8,4,7)(9,12,13,10)(11,14,15,16)(17,20,21,28)(18,23,24,30)(19,27,26,32)(22,29,25,31)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=gh,b^2=fh,c^2=gi,d^2=hi,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, 4610702);