Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
1865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,9,7)(8,11,12,10)(13,14,17,16)(15,19,20,18)(21,22)(23,27)(24,28)(25,31)(26,29)(30,35)(32,33)(34,36);(2,4)(6,7)(8,12)(21,23,26,31)(22,27,29,25)(24,32,34,30)(28,33,36,35);(1,2,3,4)(6,7)(10,11)(13,14,17,16)(15,18,20,19)(21,23)(22,25)(24,30)(26,31)(27,29)(28,33)(32,34)(35,36);(5,7,9,6)(8,11,12,10)(13,15,17,20)(14,18,16,19)(21,24,26,34)(22,28,29,36)(23,30,31,32)(25,33,27,35);(5,8)(6,10)(7,11)(9,12)(13,16,17,14)(15,19,20,18)(21,25)(22,23)(24,33)(26,27)(28,30)(29,31)(32,36)(34,35)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=fghi,d^2=gi,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1758881);