Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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:
(2,5)(6,8)(9,10,13,14)(11,12,16,15)(17,18)(19,20)(21,26)(22,23)(24,25)(27,28)(29,32)(30,31)(34,35)(36,43)(37,45)(46,47);(9,11)(10,12)(13,16)(14,15)(18,23)(19,25)(26,31)(27,29)(33,34,38,35)(36,39,43,41)(37,40,45,42)(44,46,48,47);(10,14)(11,16)(17,19)(18,24)(20,23)(21,29)(22,25)(26,28)(27,30)(31,32)(33,35,38,34)(36,39,43,41)(37,40,45,42)(44,47,48,46);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,14)(13,16)(17,20,22,24)(18,25,23,19)(21,28,30,32)(26,29,31,27)(33,36,38,43)(34,39,35,41)(37,44,45,48)(40,46,42,47);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,11,14,16)(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,32)(25,29)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=gh,d^2=f,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5453788);