Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #473461 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16866 |
| 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 | |
825 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,14,32)(3,13,27,21,17,8,18,24)(4,19,22,16,23,31,28,9)(5,10,29,15,25,30,26,20);(1,3,14,18,7,17,6,27)(2,8,11,24,12,13,32,21)(4,20,22,10,23,15,28,30)(5,16,26,19,25,9,29,31);(1,4)(2,9)(3,15)(5,24)(6,28)(7,23)(8,29)(10,27)(11,31)(12,16)(13,26)(14,22)(17,20)(18,30)(19,32)(21,25);(1,5,6,29,7,25,14,26)(2,10,11,15,12,30,32,20)(3,16,27,31,17,9,18,19)(4,21,22,8,23,24,28,13)Polycyclic presentation:
2^9|a^2=ef,b^2=fgh,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380657);