Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396717 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14737 |
| 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 | |
1161 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,13)(4,18)(5,6)(9,14)(10,25)(15,29)(16,17)(19,20)(21,24)(22,26)(23,31);(1,2,5,12,7,8,6,11)(3,14,13,21,17,24,16,9)(4,10,19,25,20,22,18,26)(15,28,27,31,29,32,30,23);(1,3)(2,9,8,21)(4,15)(5,16)(6,13)(7,17)(10,23,22,31)(11,14,12,24)(18,27)(19,30)(20,29)(25,32,26,28);(1,4)(2,10)(3,15)(5,19)(6,18)(7,20)(8,22)(9,23)(11,26)(12,25)(13,27)(14,28)(16,30)(17,29)(21,31)(24,32)Polycyclic presentation:
2^9|b^2=ef,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237066, #72237074, #72237078, … |
| maximal normal subgroup of | #72237066, #72237074, #72237078, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303913);