Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479749 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17233 |
| 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 | |
779 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)(3,14,18,29)(4,19,22,9)(5,11)(6,13)(7,12)(8,17,26,24)(10,23,27,21)(15,28)(16,25)(20,30)(31,32);(1,3,6,17,7,18,5,24)(2,8,13,14,12,26,11,29)(4,20,23,16,22,15,21,31)(9,25,10,28,19,32,27,30);(1,4,5,21,7,22,6,23)(2,9,11,27,12,19,13,10)(3,15,17,31,18,20,24,16)(8,25,14,28,26,32,29,30);(1,4)(2,10)(3,16)(5,23)(6,21)(7,22)(8,25)(9,13)(11,19)(12,27)(14,30)(15,24)(17,20)(18,31)(26,32)(28,29)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,d^c=de,g^c=gi,e^d=ei,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72251690, #72251693, #72251703, … |
| maximal normal subgroup of | #72251690, #72251693, #72251703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386945);