Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458124 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2731 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16653 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
651 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11)(3,12)(4,17,20,9)(5,7,13,10)(8,16)(14,28)(15,22)(18,26)(19,27,29,23)(21,24)(25,30)(31,32);(2,7,11,10)(8,24,12,22)(9,23,17,27)(25,31,28,26);(1,3,13,15,6,16,5,21)(2,8,7,22,11,12,10,24)(4,18,29,30,20,32,19,14)(9,26,23,28,17,31,27,25);(1,4,6,20)(2,9)(3,14)(5,19,13,29)(7,23)(8,25,12,28)(10,27)(11,17)(15,18)(16,30)(21,32)(22,26,24,31)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=efg,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^c=de,e^c=ei,f^c=fi,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72280350 |
| maximal normal subgroup of | #72280350 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376455);