Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
2100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(6,9)(7,11)(15,19)(18,20)(21,22,26,27)(23,24,31,28)(25,29,33,35)(30,32,36,34);(1,2,3,4)(7,11)(10,12)(22,27)(24,28)(25,33)(30,36);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,16)(15,18,19,20)(21,23,26,31)(22,28,27,24)(25,30,33,36)(29,34,35,32);(5,7,8,11)(6,10,9,12)(13,15,17,19)(14,18,16,20)(21,24,26,28)(22,23,27,31)(25,34,33,32)(29,36,35,30);(5,6)(7,10)(8,9)(11,12)(13,16,17,14)(15,20,19,18)(21,25)(22,29)(23,30)(24,32)(26,33)(27,35)(28,34)(31,36)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fgi,d^2=i,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,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, 3209838);