Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1964 |
| commutator | C22×C4 | C2×C42 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,16,12,7,9,14)(8,19,22,35,23,33,20,34)(10,18,25,30,26,15,24,31)(13,27,29,36,17,32,28,21);(2,4)(5,7,12,6)(8,17,23,13)(9,16)(10,18,26,15)(11,14)(19,21,33,36)(20,29)(22,28)(24,31)(25,30)(27,34)(32,35);(5,8,10,21,12,23,26,36)(6,13,15,19,7,17,18,33)(9,20,24,32,11,22,25,27)(14,28,30,34,16,29,31,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,h^a=hi,d^b=dh,f^b=fh,h^b=hi,f^d=fi,g^d=gi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72312111, #72312120, #72312136, … |
| maximal normal subgroup of | #72312111, #72312120, #72312136, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54275);