Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #473873 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16885 |
| 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 | |
914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,12,5,11)(3,13,17,25,16,24,15,8)(4,18,20,27,22,28,21,9)(14,26,30,29,19,23,31,32);(1,3,7,16)(2,8)(4,19)(5,15,6,17)(9,26,27,23)(10,13)(11,24)(12,25)(14,22)(18,29,28,32)(20,31)(21,30);(1,4)(2,9)(3,14)(5,20)(6,21)(7,22)(8,23)(10,28)(11,18)(12,27)(13,29)(15,30)(16,19)(17,31)(24,32)(25,26);(2,10)(4,20)(5,6)(8,13)(9,27)(11,12)(14,30)(15,17)(19,31)(21,22)(23,26)(24,25)Polycyclic presentation:
2^9|a^2=fg,b^2=egh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251784, #72251790, #72251803, … |
| maximal normal subgroup of | #72251784, #72251790, #72251803, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381069);