Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #466414 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16544 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
932 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)(4,7)(5,10,22,26)(6,11)(8,12)(9,16)(14,17)(15,29,32,27)(18,24)(19,21,25,20)(23,31,28,30);(1,3)(2,9,12,14)(4,13,11,24)(5,15)(6,16)(7,17)(8,18)(10,23,26,28)(19,27,25,29)(20,30)(21,31)(22,32);(1,4,6,12,8,11,7,2)(3,14,16,24,18,9,17,13)(5,10,20,25,22,26,21,19)(15,27,30,23,32,29,31,28);(1,5,7,20,8,22,6,21)(2,10,11,25,12,26,4,19)(3,15,17,30,18,32,16,31)(9,23,24,29,14,28,13,27)Polycyclic presentation:
2^9|a^2=ef,b^2=fg,c^2=e,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237067, #72237071, #72237084, … |
| maximal normal subgroup of | #72237067, #72237071, #72237084, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 373610);