Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478253 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17137 |
| 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 | |
977 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)(3,7,14,24)(4,8)(5,18)(6,10)(9,17)(11,21)(12,30,31,22)(13,23,32,27)(15,29,19,25)(16,28)(20,26);(1,3)(2,7,10,24)(4,13)(5,19)(6,14)(8,23,9,27)(11,29,18,25)(12,16)(15,21)(17,32)(20,31)(22,28,30,26);(1,4)(2,8)(3,12,14,31)(5,16)(6,17)(7,22,24,30)(9,10)(11,26)(13,19,32,15)(18,28)(20,21)(23,29,27,25);(1,4)(2,9,10,8)(3,13)(5,20)(6,17)(7,23,24,27)(11,26,18,28)(12,15)(14,32)(16,21)(19,31)(22,29,30,25)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,h^c=hi,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72285055, #72286154, #72286186, … |
| maximal normal subgroup of | #72285055, #72286154, #72286186, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385449);