Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
952 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,16,9,14)(7,17,19,12,21,27,20,28)(8,22,23,13,25,29,24,30)(18,32,34,36,35,26,33,31);(1,2,3,4)(5,7)(6,12)(8,18)(9,19)(10,20)(11,21)(13,26)(14,27)(15,17)(16,28)(22,31)(23,33)(24,34)(25,35)(29,36)(30,32);(5,8,9,23,11,25,10,24)(6,13,14,29,16,30,15,22)(7,18,19,33,21,35,20,34)(12,26,27,36,28,32,17,31);(7,19,21,20)(12,27,28,17)(18,33,35,34)(26,36,32,31)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=efg,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264258);