Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C22×#95 |
| prime radical | C25 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,8,6,7,5,3)(9,10)(11,17)(12,20)(13,16)(14,19)(15,21)(18,23)(22,24);(1,3,5,7,6,8,4,2)(9,11)(10,14,16,22)(12,19)(13,18)(15,17,20,23)(21,24);(2,7)(3,8);(11,18)(14,22)(17,23)(19,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=h,f^2=i,g^2=i,b^a=be,c^a=ci,e^a=eh,g^a=gh,c^b=ci,d^b=dh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72226705, #72226706, #72226709, … |
| maximal normal subgroup of | #72226705, #72226706, #72226709, … |
| maximal quotient of | #72239280, C2×#181445, #72309129, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88641);