Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #27987 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2504 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14020 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,10,5,11)(3,13,16,24,17,8,15,25)(4,18,21,28,20,27,22,9)(14,30,19,23,31,26,32,29);(1,3)(2,8)(4,19,20,32)(5,16)(6,15)(7,17)(9,26,28,30)(10,13)(11,25)(12,24)(14,22,31,21)(18,23,27,29);(1,4,5,21,7,20,6,22)(2,9,12,27,10,28,11,18)(3,14,16,32,17,31,15,19)(8,23,24,26,13,29,25,30);(1,3)(2,8)(4,14)(5,16)(6,15)(7,17)(9,23)(10,13)(11,25)(12,24)(18,30)(19,22)(20,31)(21,32)(26,27)(28,29)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=eg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72317961, #72318482, #72318497, … |
| maximal normal subgroup of | #72317961, #72318482, #72318497, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275746);