Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #462962 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16458 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,21,7,9,5,11)(3,12,15,24,16,22,13,8)(4,17,19,26,20,10,18,25)(14,29,30,23,31,27,28,32);(1,3)(2,8,9,24)(4,14)(5,15)(6,13)(7,16)(10,23,17,32)(11,12,21,22)(18,30)(19,28)(20,31)(25,29,26,27);(2,9)(3,13)(4,18)(5,6)(8,22)(10,25)(12,24)(15,16)(17,26)(19,20)(27,29)(28,30);(1,4,5,18,7,20,6,19)(2,10,21,25,9,17,11,26)(3,14,15,30,16,31,13,28)(8,23,22,29,24,32,12,27)Polycyclic presentation:
2^9|a^2=fg,b^2=f,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 370158);