Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #368855 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14023 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60426 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1705 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,13,8,14,7,12)(3,15,18,22,19,30,17,9)(4,20,24,10,25,31,23,32)(5,26,28,16,29,21,27,11);(1,3,6,18,8,19,7,17)(2,9,13,15,14,22,12,30)(4,21,24,11,25,26,23,16)(5,10,28,31,29,32,27,20);(1,4)(2,10,14,32)(3,16,19,11)(5,22)(6,23)(7,24)(8,25)(9,29)(12,31,13,20)(15,28)(17,21,18,26)(27,30);(1,5)(2,11)(3,10)(4,22)(6,28)(7,27)(8,29)(9,25)(12,21)(13,26)(14,16)(15,23)(17,20)(18,31)(19,32)(24,30)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72213409, #72215995, #72216029, … |
| maximal normal subgroup of | #72213409, #72215995, #72216029, … |
| maximal quotient of | C2×#368857, #72250491, #72252143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276053);