Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1958 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)(15,17)(16,19)(18,20)(21,22,25,26)(23,27,29,33)(24,28,32,34)(30,35,31,36);(5,6,9,7)(8,10,12,11)(14,15)(18,19)(22,26)(23,29)(28,34)(30,31);(5,6,9,7)(8,11,12,10)(13,15,17,14)(16,18,20,19)(21,23)(22,27)(24,31)(25,29)(26,33)(28,36)(30,32)(34,35);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(21,22)(23,27)(24,28)(25,26)(29,33)(30,35)(31,36)(32,34);(5,8,9,12)(6,10,7,11)(13,16,17,20)(14,18,15,19)(21,24,25,32)(22,28,26,34)(23,30,29,31)(27,35,33,36)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=g,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1993620);