Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(9,10)(11,17,19,24)(12,16)(13,15)(14,18,22,20)(21,23);(1,2,3,4)(5,6)(7,8);(6,8)(9,11)(10,14)(12,20)(13,19)(15,22)(16,24)(17,23)(18,21);(10,15)(14,22)(16,23)(17,24)Polycyclic presentation:
2^9|a^2=h,b^2=e,f^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,f^c=fi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72237203, #72237228, #72237245, … |
| maximal normal subgroup of | #72237203, #72237228, #72237245, … |
| maximal quotient of | #72238765, #72238768, #72239438, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264849);