Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478920 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17175 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
821 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13)(4,9,21,28)(5,11)(6,12)(7,10)(8,16)(14,29,30,25)(15,23)(17,24)(18,32,31,22)(19,27,20,26);(1,3,7,16,6,17,5,15)(2,8,10,24,12,23,11,13)(4,18,19,30,21,31,20,14)(9,25,27,22,28,29,26,32);(1,4,7,19,6,21,5,20)(2,9,10,27,12,28,11,26)(3,14,16,18,17,30,15,31)(8,22,24,29,23,32,13,25);(3,15)(4,19)(5,7)(8,13)(9,26)(10,11)(14,30)(16,17)(20,21)(23,24)(25,29)(27,28)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,c^2=ef,e^2=i,g^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=def,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei,g^d=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72251688, #72251693, #72251703, … |
| maximal normal subgroup of | #72251688, #72251693, #72251703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386116);