Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417622 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1969 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,22)(9,14)(10,15)(11,16)(12,20)(13,24)(18,30)(19,28)(21,29)(23,27)(25,32)(26,31)(33,35)(34,36);(5,7,10,19,11,21,9,20)(6,12,14,29,16,28,15,17)(8,23,24,34,26,18,25,35)(13,30,22,36,32,27,31,33);(5,8)(6,13)(7,18)(9,24)(10,25)(11,26)(12,27)(14,31)(15,22)(16,32)(17,33)(19,34)(20,35)(21,23)(28,30)(29,36);(2,4)(6,14)(7,19)(9,10)(12,28)(13,31)(15,16)(18,34)(20,21)(22,32)(23,35)(24,25)(27,30)Polycyclic presentation:
2^9|b^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328037);