Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,4,7)(11,15)(14,16)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32)(33,34)(35,40)(36,37)(38,41)(39,42)(43,47)(44,45)(46,48);(10,13)(14,16)(18,22)(19,26)(24,28)(27,32)(33,35,39,36)(34,40,42,37)(38,44,46,43)(41,45,48,47);(1,3,5,4)(2,6,8,7)(9,10,12,13)(11,14,15,16)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32)(33,36,39,35)(34,40,42,37)(38,44,46,43)(41,47,48,45);(1,2)(3,6)(4,7)(5,8)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32)(33,37,39,40)(34,36,42,35)(38,45,46,47)(41,44,48,43);(1,4,5,3)(2,7,8,6)(9,11,12,15)(10,14,13,16)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)(33,38,39,46)(34,41,42,48)(35,43,36,44)(37,45,40,47)Polycyclic presentation:
2^9|a^2=fi,b^2=g,c^2=ghi,d^2=g,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5369644);