Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C2×#3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6,7,8)(9,10)(11,17,19,14)(12,16)(13,15)(18,23,24,21)(20,22);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23);(10,15)(14,17)(16,22)(21,23);(1,3,5,4)(2,6,7,8);(9,12,13,20)(10,16,15,22)(11,18,19,24)(14,21,17,23)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=ch,f^a=fh,f^b=fi,g^b=ghi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72239188, #72239232, #72239238, … |
| maximal normal subgroup of | #72239188, #72239232, #72239238, … |
| maximal quotient of | #72239193, #72239262, #72239593, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250498);