Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | |
1354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,3)(4,6,8,7)(9,10,13,11)(12,15,16,14)(18,22)(19,25)(24,29)(27,31)(33,34,37,38)(35,41,43,39)(36,44,45,40)(42,46,48,47);(4,8)(6,7)(9,10,13,11)(12,15,16,14)(18,22)(19,25)(20,28)(30,32)(34,38)(39,41)(40,44)(46,47);(2,3)(6,7)(10,11)(14,15)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,35,37,43)(34,39,38,41)(36,42,45,48)(40,46,44,47);(1,3,5,2)(4,7,8,6)(9,11,13,10)(12,15,16,14)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,36,37,45)(34,40,38,44)(35,42,43,48)(39,46,41,47);(1,4,5,8)(2,6,3,7)(9,12,13,16)(10,14,11,15)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=fhi,c^2=i,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6189392);