Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #436179 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15550 |
| 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 | |
952 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)(3,4,14,18)(5,9)(6,10)(7,11)(8,13,17,22)(12,16,27,30)(15,23)(19,29,31,24)(20,25)(21,26)(28,32);(1,3)(2,8)(4,15)(5,19)(6,22)(7,14)(9,16)(10,18)(11,17)(12,20)(13,23)(21,31)(24,25)(26,30)(27,32)(28,29);(1,4,6,17,7,18,23,8)(2,3,10,13,11,14,15,22)(5,16,20,29,21,30,32,24)(9,12,25,19,26,27,28,31);(1,5)(2,9,11,26)(3,12,14,27)(4,16)(6,20)(7,21)(8,24)(10,25,15,28)(13,19,22,31)(17,29)(18,30)(23,32)Polycyclic presentation:
2^9|a^2=fg,c^2=e,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72338616, #72338621, #72338632 |
| maximal normal subgroup of | #72338616, #72338621, #72338632 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343375);