Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31077 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2804 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,9,15,11,16,10,14)(7,17,19,29,21,28,20,12)(8,22,25,31,24,32,26,13)(18,34,35,30,36,27,23,33);(1,2)(3,4)(5,7)(6,12)(8,23)(9,19)(10,20)(11,21)(13,30)(14,28)(15,17)(16,29)(18,25)(22,27)(24,35)(26,36)(31,33)(32,34);(5,8)(6,13,16,31)(7,18,21,36)(9,25)(10,26)(11,24)(12,27)(14,32,15,22)(17,33)(19,35,20,23)(28,30)(29,34);(1,3,4,2)(7,19,21,20)(8,24)(12,28,29,17)(13,31)(18,23,36,35)(22,32)(25,26)(27,33,34,30)Polycyclic presentation:
2^9|a^2=h,c^2=fg,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214494, #72215059, #72216206, … |
| maximal normal subgroup of | #72214494, #72215059, #72216206, … |
| maximal quotient of | #72214172, #72214180, #72214944, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293999);