Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #424740 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15324 |
| 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 | |
1106 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)(3,13)(4,10,19,25)(5,8)(6,11)(7,12)(9,17)(14,23)(15,27,28,32)(16,24)(18,26,20,21)(22,29,31,30);(2,8)(3,14)(4,18)(5,6)(9,13)(10,25)(11,12)(15,29)(19,20)(22,31)(23,24)(28,30);(1,3,6,17,7,14,5,16)(2,9,8,13,12,24,11,23)(4,15,18,30,19,28,20,29)(10,22,21,27,25,31,26,32);(1,4)(2,10)(3,15)(5,20)(6,18)(7,19)(8,21)(9,22)(11,26)(12,25)(13,27)(14,28)(16,29)(17,30)(23,32)(24,31)Polycyclic presentation:
2^9|a^2=ef,c^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
| maximal subgroup of | #72251784, #72251789, #72251799, … |
| maximal normal subgroup of | #72251784, #72251789, #72251799, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331936);