Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1180 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)(6,9)(7,11)(21,22)(23,26)(24,27)(25,29)(28,32)(30,35)(31,34)(33,36);(5,6,8,9)(7,12,11,10)(13,14,16,18)(15,19,20,17)(21,23,25,30)(22,26,29,35)(24,31,32,36)(27,34,28,33);(13,15,16,20)(14,17,18,19)(22,27)(23,30)(24,32)(26,33)(28,29)(34,35);(5,7,8,11)(6,10,9,12)(22,27)(24,32)(26,34)(28,29)(31,36)(33,35)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=gi,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,f^c=fi,f^d=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294308);