Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #361316 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13894 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1431 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)(3,7,12,22)(4,14,17,8)(5,10)(6,9)(11,29)(13,23,18,27)(15,32)(16,28,19,25)(20,26)(21,30)(24,31);(1,3)(2,7,9,22)(4,15)(5,18)(6,12)(8,24,14,32)(10,27,26,23)(11,16)(13,20)(17,31)(19,30)(21,28,29,25);(1,4)(2,8)(3,11)(5,19)(6,17)(7,21)(9,14)(10,28)(12,30)(13,31)(15,18)(16,20)(22,29)(23,32)(24,27)(25,26);(1,3)(2,7,9,22)(4,11)(5,13)(6,12)(8,21,14,29)(10,23,26,27)(15,16)(17,30)(18,20)(19,31)(24,28,32,25)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72237642, #72240432, #72240447, … |
| maximal normal subgroup of | #72237642, #72240432, #72240447, … |
| maximal quotient of | #72237662, C2×#361316 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268512);