Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | D4×C4⋊C4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,17)(7,18)(8,22,25,30)(9,12)(10,15,26,16)(13,28,31,24)(14,20)(19,29)(21,32)(23,33)(27,35)(34,36);(1,2)(3,4)(5,7)(6,12)(8,19)(9,10)(11,21)(13,27)(14,17)(15,32)(16,18)(20,26)(22,33)(23,24)(25,35)(28,34)(29,31)(30,36);(2,4)(5,8,10,24,11,25,26,28)(6,13,16,22,17,31,15,30)(7,19,9,23,21,35,20,34)(12,27,18,33,14,29,32,36);(5,9,10,21,11,20,26,7)(6,14,16,32,17,12,15,18)(8,23,24,35,25,34,28,19)(13,29,22,36,31,27,30,33)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=ei,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292385);