Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
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 | |
1870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,10,11)(7,8,9,12)(21,22,25,28)(23,26,29,33)(24,31,32,27)(30,36,35,34);(1,2,3,4)(6,11)(7,9)(14,17)(18,20)(23,29)(26,33)(30,35)(34,36);(5,7,10,9)(6,12,11,8)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,32)(35,36);(5,8)(6,7)(9,11)(10,12)(13,15,16,19)(14,18,17,20)(21,23,25,29)(22,26,28,33)(24,30,32,35)(27,34,31,36);(5,9,10,7)(6,8,11,12)(21,24,25,32)(22,27,28,31)(23,30,29,35)(26,34,33,36)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=f,d^2=gh,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=ef104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 467218);