Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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 | |
1918 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)(7,11)(8,10)(9,12)(13,14)(15,20)(16,17)(18,19)(21,22,25,28)(23,26,29,32)(24,27,31,34)(30,33,35,36);(1,2,4,3)(5,6,8,10)(7,9,12,11)(14,19)(15,17)(23,29)(26,32)(30,35)(33,36);(1,3,4,2)(13,15,18,17)(14,20,19,16)(21,22)(23,26)(24,27)(25,28)(29,32)(30,33)(31,34)(35,36);(5,7)(6,9)(8,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,23,25,29)(22,26,28,32)(24,30,31,35)(27,33,34,36);(5,6)(7,9)(8,10)(11,12)(13,17,18,15)(14,16,19,20)(21,24,25,31)(22,27,28,34)(23,30,29,35)(26,33,32,36)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fi,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=dg,e^b=ef104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 463524);