Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,4)(6,7)(9,10,13,11)(12,15,16,14)(18,22)(19,25)(24,29)(27,31)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,47)(45,46);(1,2,5,8)(3,6,4,7)(9,10,13,11)(12,14,16,15)(20,28)(24,29)(27,31)(30,32)(34,38)(39,41)(40,46)(47,48);(1,2)(3,6)(4,7)(5,8)(9,11,13,10)(12,15,16,14)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,47)(45,46);(1,3,5,4)(2,6,8,7)(9,12)(10,14)(11,15)(13,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,35,37,43)(34,39,38,41)(36,44,45,42)(40,48,46,47);(1,4,5,3)(2,7,8,6)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,36,37,45)(34,40,38,46)(35,42,43,44)(39,47,41,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=fgi,c^2=ghi,d^2=h,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5717637);