Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,7)(3,4)(11,15)(14,16)(18,22)(20,27)(23,28)(26,31)(33,34,38,39)(35,42,41,37)(36,40,44,47)(43,45,48,46);(1,2)(3,8)(4,5)(6,7)(9,10,13,12)(11,14,15,16)(18,22)(19,25)(24,29)(26,31)(34,39)(35,41)(36,44)(45,46);(1,3,6,4)(2,8,7,5)(9,10,13,12)(11,14,15,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,38,41)(34,37,39,42)(36,43,44,48)(40,46,47,45);(1,4,6,3)(2,8,7,5)(9,11)(10,14)(12,16)(13,15)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,36)(34,40)(35,43)(37,46)(38,44)(39,47)(41,48)(42,45);(1,5,6,8)(2,4,7,3)(9,12,13,10)(11,16,15,14)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37)(34,41)(35,39)(36,45)(38,42)(40,43)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=fgh,b^2=ghi,c^2=hi,d^2=fh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4835749);