Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1999 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)(6,7)(8,12)(13,14)(15,18)(16,20)(17,19)(21,22,25,26)(23,29,30,27)(24,28,33,32)(31,35,36,34);(6,7)(10,11)(13,15,17,16)(14,18,19,20)(22,26)(23,30)(28,32)(31,36);(5,6,9,7)(8,10,12,11)(13,16,17,15)(14,18,19,20)(21,22)(23,27)(24,32)(25,26)(28,33)(29,30)(31,35)(34,36);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,20)(16,18)(17,19)(21,23,25,30)(22,27,26,29)(24,31,33,36)(28,34,32,35);(5,8)(6,10)(7,11)(9,12)(21,24,25,33)(22,28,26,32)(23,31,30,36)(27,34,29,35)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=fg,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4507873);