Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417901 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2679 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1025 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,19)(12,15)(13,17)(14,18)(16,21)(20,24)(22,23);(1,3)(2,6)(4,7)(5,8)(10,15,18,17)(16,19,24,23);(1,4,5,2)(3,7,8,6)(9,11)(10,16)(12,21)(13,20)(14,22)(15,23)(17,19)(18,24);(3,8)(6,7)(9,10)(11,16)(12,15)(13,17)(14,18)(19,20)(21,23)(22,24)Polycyclic presentation:
2^9|a^2=ef,b^2=g,c^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=dei,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72339028, #72339040, #72339072, … |
| maximal normal subgroup of | #72339028, #72339040, #72339072, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 345821);