Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | |
1298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(17,18,21,22)(19,23,27,25)(20,24,29,30)(26,31,28,32)(33,34)(35,40)(36,41)(37,45)(38,39)(42,47)(43,44)(46,48);(1,2,5,4)(3,6,7,8)(10,13)(14,16)(18,22)(23,25)(24,30)(31,32)(34,39)(36,40)(42,45)(44,46);(9,10,12,13)(11,14,15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,31)(29,30)(33,35)(34,40)(36,39)(37,43)(38,41)(42,46)(44,45)(47,48);(1,3)(2,6)(4,8)(5,7)(17,19)(18,23)(20,28)(21,27)(22,25)(24,32)(26,29)(30,31)(33,36,38,40)(34,41,39,35)(37,46,47,44)(42,43,45,48);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)(17,20,21,29)(18,24,22,30)(19,26,27,28)(23,31,25,32)(33,37,38,47)(34,42,39,45)(35,43,41,48)(36,44,40,46)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,d^2=fg,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2707553);