Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135286 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1823 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,16,27,35,9,15,12,18,26,14,11,17,8,25)(7,19,22,13,29,31,20,36,24,30,28,33,23,34,21,32);(1,2)(3,4)(5,7)(6,13,18,33)(8,22)(9,28)(10,21)(11,29)(12,24)(14,30,16,19)(15,34,25,31)(17,36,35,32)(20,26)(23,27);(2,4)(6,14,18,16)(8,26)(9,10)(11,27)(13,30,33,19)(15,17,25,35)(20,22)(21,28)(23,29)(31,32,34,36)Polycyclic presentation:
2^9|a^2=g,b^2=deh,c^2=deh,d^2=h,e^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=dh,e^b=eh,g^b=gh,h^b=hi,d^c=dh,e^c=eh,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224839, #72224931, #72226819, … |
| maximal normal subgroup of | #72224839, #72224931, #72226819, … |
| maximal quotient of | #72224832 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56571);