Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457100 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16290 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23439 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,24,7,12,22,11)(3,13,15,8,17,28,16,27)(4,18,5,21,20,9,23,10)(14,30,31,25,32,29,19,26);(1,3)(2,8,12,27)(4,19)(5,14)(6,15)(7,17)(9,29,18,30)(10,26,21,25)(11,13,24,28)(16,22)(20,31)(23,32);(1,4)(2,9)(3,14)(5,22)(6,23)(7,20)(8,25)(10,11)(12,18)(13,29)(15,19)(16,31)(17,32)(21,24)(26,27)(28,30);(1,5,22,4,7,23,6,20)(2,10,24,18,12,21,11,9)(3,14,16,19,17,32,15,31)(8,26,28,30,27,25,13,29)Polycyclic presentation:
2^9|a^2=ef,b^2=f,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364297);