Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #482075 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#3169 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
907 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(9,10,12,15,14,18,11,19,13,16,20,23,21,24,22,17);(1,2)(3,6)(4,8)(5,7)(10,15)(11,20)(12,22)(14,21)(16,23)(17,18)(19,24);(1,3,5,4)(2,6,7,8)(10,16)(15,23)(17,19)(18,24);(10,16)(15,23)(17,19)(18,24)Polycyclic presentation:
2^9|a^2=f,c^2=gi,e^2=hi,f^2=ghi,h^2=i,b^a=be,c^a=cg,d^a=di,e^a=eg,e^b=eh,f^b=fgh,h^b=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72309626, #72309639, #72309658, … |
| maximal normal subgroup of | #72309626, #72309639, #72309658, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390374);