Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99923 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
916 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,10)(8,15,16,11)(9,12)(13,17,20,19)(14,18)(21,22,24,28,25,29,33,27)(23,26,30,34,32,36,31,35);(1,2,3,4)(5,7)(6,10)(8,13)(9,14)(11,19)(12,18)(15,17)(16,20)(21,23)(22,26)(24,30)(25,32)(27,35)(28,34)(29,36)(31,33);(5,8)(6,11)(7,13)(9,16)(10,17)(12,15)(14,20)(18,19)(22,27)(23,30)(24,33)(28,29)(31,32)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fh,c^b=cf,e^b=eh,f^b=fi,e^c=ei,f^c=fi,g^c=ghi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23137);