Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #449115 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15931 |
| 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 | |
750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,11,6,12,7,16)(3,4,14,19,15,9,24,18)(5,20,22,26,21,10,23,27)(13,17,29,31,30,25,28,32);(1,3,7,14,6,15,8,24)(2,9,11,19,12,4,16,18)(5,13,22,28,21,30,23,29)(10,17,26,31,20,25,27,32);(1,4)(2,3,12,15)(5,17)(6,9)(7,18)(8,19)(10,13,20,30)(11,14,16,24)(21,25)(22,31)(23,32)(26,28,27,29);(1,5,7,23,6,21,8,22)(2,10,11,27,12,20,16,26)(3,13,14,29,15,30,24,28)(4,17,18,32,9,25,19,31)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=ei,d^2=eg,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72319143, #72319370, #72319453, … |
| maximal normal subgroup of | #72319143, #72319370, #72319453, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356312);