Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23906 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1602 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10,13,17,14,18,12,21)(11,15,19,16,20,24,22,23);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,22)(13,19)(14,20)(16,17)(18,24)(21,23);(1,4,5,3)(2,6,7,8)(9,11)(10,16)(12,22)(13,19)(14,20)(15,21)(17,24)(18,23);(2,7)(6,8)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72278275, #72278354, #72278601, … |
| maximal normal subgroup of | #72278275, #72278354, #72278601, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267137);