Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,16,11,15,9,14)(7,12,18,26,19,27,20,30)(8,21,24,29,23,28,22,13)(17,31,32,36,34,35,33,25);(5,7)(6,12,15,27)(8,17)(9,18)(10,20)(11,19)(13,25,29,36)(14,26,16,30)(21,35,28,31)(22,32)(23,34)(24,33);(1,2,3,4)(5,8)(6,13,15,29)(7,17)(9,22)(10,24)(11,23)(12,25,27,36)(14,28,16,21)(18,32)(19,34)(20,33)(26,31,30,35);(5,9,11,10)(6,14,15,16)(7,18,19,20)(8,22,23,24)(12,26,27,30)(13,28,29,21)(17,32,34,33)(25,31,36,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,e^b=ei,f^b=fi,g^b=gi,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72239305, #72239774, #72239777, … |
| maximal normal subgroup of | #72239305, #72239774, #72239777, … |
| maximal quotient of | #72239188, #72239190, #72239191, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261320);