Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| 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 | |
1188 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)(7,10)(8,12)(9,11)(21,22)(23,30)(24,27)(25,29)(26,32)(28,34)(31,36)(33,35);(5,7,9,8)(6,10,11,12)(13,14,16,18)(15,19,20,17)(21,23,25,31)(22,26,29,35)(24,33,34,32)(27,30,28,36);(6,11)(10,12)(13,15,16,20)(14,17,18,19)(23,31)(26,35)(30,36)(32,33);(5,8,9,7)(6,12,11,10)(22,27)(23,31)(24,34)(26,36)(28,29)(30,35)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=gi,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 270266);