Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(6,7)(8,12)(15,19)(17,20)(21,22)(23,29)(24,32)(25,26)(27,30)(28,33)(31,35)(34,36);(6,7)(10,11)(13,14,16,18)(15,17,19,20)(22,25)(23,30)(24,33)(35,36);(1,2,4,3)(5,6,9,7)(8,10,12,11)(13,14)(15,17)(16,18)(19,20)(21,23)(22,27)(24,34)(25,29)(26,30)(28,36)(31,33)(32,35);(1,3,4,2)(5,7,9,6)(8,11,12,10)(13,15)(14,17)(16,19)(18,20)(21,24)(22,28)(23,31)(25,32)(26,33)(27,35)(29,36)(30,34);(5,8,9,12)(6,10,7,11)(21,25,26,22)(23,29,30,27)(24,32,33,28)(31,36,34,35)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=fgi,d^2=fgi,e^2=g,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 503321);