Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #389324 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14539 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
985 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,6,10,20,9)(3,7,15,22,13,24,14,23)(4,16,19,8,18,26,32,27)(12,28,17,31,29,25,30,21);(1,3,5,14,6,13,20,15)(2,7,9,22,10,24,11,23)(4,17,19,12,18,30,32,29)(8,25,16,21,27,28,26,31);(1,4,6,18)(2,8)(3,12)(5,19,20,32)(7,21,24,31)(9,16)(10,27)(11,26)(13,29)(14,30)(15,17)(22,28,23,25);(2,9)(3,13)(5,20)(7,22)(8,16)(10,11)(12,29)(19,32)(21,28)(23,24)(25,31)(26,27)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fh,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^d=fi,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72251685, #72251692, #72251708, … |
| maximal normal subgroup of | #72251685, #72251692, #72251708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296522);