Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1388 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)(17,18,22,19)(20,25,27,23)(21,24,30,26)(28,32,29,31)(33,34)(35,36)(37,45)(38,39)(40,42)(41,48)(43,47)(44,46);(2,5)(6,8)(9,10,13,11)(12,14,16,15)(18,19)(20,27)(24,26)(28,29)(34,39)(35,42)(41,45)(43,46);(1,2,4,5)(3,6,7,8)(9,11,13,10)(12,15,16,14)(17,19,22,18)(20,25,27,23)(21,24,30,26)(28,31,29,32)(33,35)(34,40)(36,39)(37,46)(38,42)(41,44)(43,48)(45,47);(1,3)(2,6)(4,7)(5,8)(9,12,13,16)(10,14,11,15)(17,20)(18,23)(19,25)(21,29)(22,27)(24,32)(26,31)(28,30)(33,36,38,40)(34,35,39,42)(37,47,48,44)(41,46,45,43);(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,37,38,48)(34,41,39,45)(35,43,42,46)(36,44,40,47)Polycyclic presentation:
2^9|a^2=fg,b^2=ghi,c^2=i,d^2=hi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3475914);