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) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,17,18)(15,19,20,16)(21,22)(23,27)(24,31)(25,26)(28,32)(29,35)(30,33)(34,36);(8,12)(10,11)(13,15)(14,16)(17,20)(18,19)(22,25)(27,33)(28,31)(34,35);(6,7)(10,11)(13,15,17,20)(14,16,18,19)(21,23,26,30)(22,27,25,33)(24,29,32,36)(28,34,31,35);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,16)(17,20)(18,19)(21,24)(22,28)(23,29)(25,31)(26,32)(27,34)(30,36)(33,35);(5,8)(6,10)(7,11)(9,12)(13,16)(14,15)(17,19)(18,20)(21,25,26,22)(23,27,30,33)(24,31,32,28)(29,34,36,35)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=fi,e^2=fh,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, 5416911);