Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×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 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
942 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)(5,6,9,8)(7,11,12,10)(13,14)(15,18)(16,20)(17,19)(21,22,28,34)(23,35,24,31)(25,30,36,29)(26,33,27,32);(5,7)(6,10)(8,11)(9,12)(13,15,17,16)(14,18,19,20)(21,23)(22,29)(24,28)(25,26)(27,36)(30,34)(31,32)(33,35);(5,8,9,6)(7,11,12,10)(14,19)(18,20)(21,24,27,36,28,23,26,25)(22,30,33,35,34,29,32,31);(2,4)(7,12)(10,11)(13,16,17,15)(14,20,19,18)(21,25,27,24,28,36,26,23)(22,31,33,30,34,35,32,29)Polycyclic presentation:
2^9|a^2=eg,b^2=fgi,c^2=ei,d^2=fi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 349756);