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 | D4×C4⋊C4 |
| 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:
(1,2,4,3)(6,12,15,16)(7,17,19,24)(14,20,29,30)(18,31,33,34);(5,6,9,16,11,15,10,12)(7,13,17,27,19,28,24,25)(8,20,21,14,23,30,22,29)(18,32,31,26,33,35,34,36);(1,2,4,3)(5,7)(6,13)(8,18)(9,17)(10,24)(11,19)(12,25)(14,26)(15,28)(16,27)(20,32)(21,31)(22,34)(23,33)(29,36)(30,35);(1,3,4,2)(5,8,10,22,11,23,9,21)(6,14,16,30,15,29,12,20)(7,18,24,34,19,33,17,31)(13,26,27,35,28,36,25,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=efg,d^2=fg,e^2=i,f^2=i,h^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, 264240);