Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #452537 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16051 |
| 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 | |
925 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,13)(3,9)(4,16,19,10)(5,21)(7,23,8,12)(11,15)(14,30)(17,27)(18,25)(20,28,24,29)(22,26)(31,32);(1,3)(2,9)(4,17,19,31)(5,6)(7,15)(8,22)(10,27,16,32)(11,23)(12,26)(13,21)(14,20,18,24)(25,28,30,29);(1,4,6,19)(2,10)(3,14,5,18)(7,24,8,20)(9,25)(11,27)(12,29)(13,16)(15,31,22,17)(21,30)(23,28)(26,32);(1,5,6,3)(2,11)(4,18,19,14)(7,15,8,22)(9,23)(10,27)(12,21)(13,26)(16,32)(17,24,31,20)(25,28)(29,30)Polycyclic presentation:
2^9|a^2=egh,b^2=f,c^2=ef,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237069, #72237073, #72237084, … |
| maximal normal subgroup of | #72237069, #72237073, #72237084, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 359733);