Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479675 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17224 |
| 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 | |
873 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)(3,7,13,23)(4,15)(5,9)(6,10)(8,16)(11,20)(12,21,28,25)(14,22,18,24)(17,27)(19,26)(29,32,30,31);(1,3)(2,7,10,23)(4,12)(5,18)(6,13)(8,25,15,21)(9,24,11,22)(14,20)(16,28)(17,30)(19,29)(26,32,27,31);(1,4)(2,8,10,15)(3,12)(5,19)(6,16)(7,21,23,25)(9,27,11,26)(13,28)(14,30)(17,20)(18,29)(22,31,24,32);(2,9)(5,20)(7,22)(8,26)(10,11)(12,28)(14,18)(15,27)(17,19)(21,31)(23,24)(25,32)Polycyclic presentation:
2^9|a^2=eg,b^2=g,c^2=g,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,h^c=hi,h^d=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72285055, #72285150, #72286082, … |
| maximal normal subgroup of | #72285055, #72285150, #72286082, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386871);