Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393909 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(21,22,26,30)(23,31,32,35)(24,29,25,28)(27,34,36,33);(2,4)(5,7,9,8)(6,10,11,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,27)(24,34)(25,33)(26,32)(28,35)(29,31)(30,36);(2,4)(5,8,9,7)(6,10,11,12)(14,17)(18,20)(22,28)(23,32)(24,25)(27,31)(29,30)(35,36);(13,15,16,19)(14,18,17,20)(22,29)(24,25)(27,31)(28,30)(33,34)(35,36)Polycyclic presentation:
2^9|a^2=i,b^2=ef,c^2=ef,d^2=gi,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,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312068);