Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #432901 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15468 |
| 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 | |
1018 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)(3,8,13,4)(5,17)(6,21)(7,10)(9,20)(11,25)(12,29,18,32)(14,16,22,23)(15,30,26,31)(19,28)(24,27);(1,3)(2,8,10,4)(5,18,19,12)(6,22)(7,13)(9,26)(11,16,21,23)(14,25)(15,27)(17,32)(20,30,24,31)(28,29);(1,4,7,8)(2,3)(5,15,19,26)(6,23,25,16)(9,12)(10,13)(11,22)(14,21)(17,31)(18,27)(20,29,24,32)(28,30);(1,5)(2,9)(3,12)(4,15)(6,24)(7,19)(8,26)(10,27)(11,28)(13,18)(14,31)(16,29)(17,21)(20,25)(22,30)(23,32)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=ei,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,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72285150, #72286154, #72286186, … |
| maximal normal subgroup of | #72285150, #72286154, #72286186, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340098);