Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D8 |
| commutator | C8 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,13,26,16)(7,17,19,29,21,34,20,36)(8,14,22,32,23,31,9,24)(12,27,30,35,28,25,33,18);(1,2,4,3)(5,7)(6,12)(8,18)(9,25)(10,19)(11,21)(13,28)(14,29)(15,30)(16,33)(17,24)(20,26)(22,27)(23,35)(31,36)(32,34);(1,3,4,2)(6,13)(12,28)(14,31)(15,16)(17,34)(24,32)(29,36)(30,33);(5,8,10,22,11,23,26,9)(6,14,15,32,13,31,16,24)(7,18,19,27,21,35,20,25)(12,29,30,34,28,36,33,17)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=f,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=ci,e^a=eh,h^a=hi,e^b=eh,h^b=hi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72216211, #72216219, #72216223, … |
| maximal normal subgroup of | #72216211, #72216219, #72216223, … |
| maximal quotient of | C8×#2177, C2×#493242, C8×#2179 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400438);