Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #449117 |
| 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 | |
765 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,11,6,13,7,12)(3,4,15,19,16,9,24,18)(5,20,22,10,21,26,23,27)(14,28,30,17,31,32,29,25);(1,3,7,15,6,16,8,24)(2,9,11,19,13,4,12,18)(5,14,22,29,21,31,23,30)(10,17,20,32,27,25,26,28);(1,4)(2,3,13,16)(5,17)(6,9)(7,18)(8,19)(10,14,27,31)(11,15,12,24)(20,29,26,30)(21,25)(22,32)(23,28);(1,5,7,23,6,21,8,22)(2,10,11,26,13,27,12,20)(3,14,15,30,16,31,24,29)(4,17,18,28,9,25,19,32)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=ei,d^2=eg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237066, #72237073, #72237084, … |
| maximal normal subgroup of | #72237066, #72237073, #72237084, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356314);