Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| 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 | |
900 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)(5,6,8,10)(7,9,11,12)(14,17)(18,20)(21,22,26,31)(23,32,35,36)(24,30,25,29)(27,34,28,33);(7,11)(9,12)(13,14)(15,18)(16,17)(19,20)(21,23,24,33,26,35,25,34)(22,27,29,32,31,28,30,36);(2,4)(5,7,8,11)(6,9,10,12)(14,17)(15,19)(23,33)(24,25)(27,32)(28,36)(29,30)(34,35);(13,15,16,19)(14,18,17,20)(21,23,25,34,26,35,24,33)(22,28,30,32,31,27,29,36)Polycyclic presentation:
2^9|a^2=egi,b^2=hi,c^2=eg,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381621);