Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135848 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1853 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(6,12,15,28)(8,21)(9,10)(13,26,32,31)(14,29,16,27)(17,34,33,36)(18,35)(19,25)(22,30)(23,24);(5,6,8,12,10,16,21,29,11,15,24,28,9,14,23,27)(7,17,18,34,19,32,35,26,20,33,22,36,25,13,30,31);(1,2,3,4)(5,7)(6,13)(8,22)(9,25)(10,19)(11,20)(12,26)(14,33)(15,32)(16,17)(18,24)(21,30)(23,35)(27,34)(28,31)(29,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,d^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,g^b=gi,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224903, #72224939, #72224942, … |
| maximal normal subgroup of | #72224903, #72224939, #72224942, … |
| maximal quotient of | #72224897, #72237232, C2×#149198 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56394);