Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
1294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(9,10,13,11)(12,15,16,14)(17,18,22,23)(19,24,25,20)(21,26,29,30)(27,31,32,28)(34,35)(37,45)(39,41)(43,48);(2,6)(7,8)(18,23)(19,25)(21,29)(28,31)(33,34,38,35)(36,41,44,39)(37,42,45,40)(43,46,48,47);(1,2)(3,7)(4,8)(5,6)(9,11,13,10)(12,15,16,14)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32)(33,35,38,34)(36,39,44,41)(37,42,45,40)(43,46,48,47);(1,3,5,4)(2,7,6,8)(9,12,13,16)(10,14,11,15)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,4,5,3)(2,8,6,7)(17,21,22,29)(18,26,23,30)(19,27,25,32)(20,28,24,31)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fi,d^2=fhi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1687759);