Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476849 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17098 |
| 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 | |
901 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,19,6,7,20,10)(3,11,14,8,15,24,12,21)(4,9,16,27,18,22,17,25)(13,28,29,26,30,32,31,23);(2,7)(3,12)(4,16)(5,20)(8,11)(9,25)(13,30)(14,15)(17,18)(21,24)(22,27)(23,26);(1,3)(2,8,7,21)(4,13)(5,14)(6,15)(9,26,22,23)(10,11,19,24)(12,20)(16,29)(17,31)(18,30)(25,28,27,32);(1,4)(2,9,7,22)(3,13)(5,17)(6,18)(8,23,21,26)(10,27,19,25)(11,28,24,32)(12,29)(14,31)(15,30)(16,20)Polycyclic presentation:
2^9|a^2=ef,c^2=e,d^2=e,f^2=i,g^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=def,f^b=fi,g^b=gi,d^c=de,g^c=gi,f^d=fi,h^d=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72237067, #72237068, #72237086, … |
| maximal normal subgroup of | #72237067, #72237068, #72237086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384045);