Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C2×#3460 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,13)(12,16)(14,18)(15,17)(19,20)(21,22,24,27,25,28,31,34)(23,26,30,33,29,32,35,36);(6,10)(8,16)(11,17)(12,13)(15,20)(18,19)(21,23)(22,26)(24,30)(25,29)(27,33)(28,32)(31,35)(34,36);(2,4)(5,7)(6,11)(8,15)(9,14)(10,17)(12,19)(13,18)(16,20)(23,29)(26,32)(30,35)(33,36);(7,14)(11,18)(15,20)(17,19)Polycyclic presentation:
2^9|a^2=g,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ci,e^b=eh,d^c=dh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72320684, #72320715, #72320826 |
| maximal normal subgroup of | #72320684, #72320715, #72320826 |
| maximal quotient of | #72321250 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92053);