Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| 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 | |
2152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(5,6,8,10)(7,11,12,9)(14,18)(15,16)(21,22)(23,30)(24,28)(25,33)(26,27)(29,34)(31,32)(35,36);(14,18)(19,20)(22,27)(23,28)(25,34)(32,36);(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,20)(17,18)(21,23)(22,24)(25,31)(26,28)(27,30)(29,32)(33,36)(34,35);(5,6)(7,9)(8,10)(11,12)(13,15,17,16)(14,19,18,20)(21,24)(22,28)(23,27)(25,32)(26,30)(29,35)(31,33)(34,36);(1,3,4,2)(5,7,8,12)(6,9,10,11)(13,16,17,15)(14,20,18,19)(21,25)(22,29)(23,31)(24,32)(26,34)(27,33)(28,35)(30,36)Polycyclic presentation:
2^9|a^2=fhi,c^2=fgi,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1082394);