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 | |
1370 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)(3,4)(5,6)(7,8)(9,10)(11,16)(12,15)(13,14)(18,22)(20,27)(23,28)(26,31)(33,34,38,37)(35,41,43,39)(36,44,45,40)(42,46,48,47);(2,6)(3,8)(9,10,14,13)(11,15,12,16)(18,22)(19,25)(24,29)(26,31)(36,45)(40,44)(42,48)(46,47);(2,6)(4,7)(9,11,14,12)(10,15,13,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35,38,43)(34,39,37,41)(36,42,45,48)(40,46,44,47);(1,3,5,8)(2,7,6,4)(9,12,14,11)(10,15,13,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48);(1,4,5,7)(2,8,6,3)(9,13,14,10)(11,15,12,16)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,34)(35,41,43,39)(36,44,45,40)(42,47,48,46)Polycyclic presentation:
2^9|a^2=fgh,b^2=ghi,c^2=fi,d^2=fh,e^2=gh,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, 4571209);