Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93073 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1296 |
| commutator | C4×C8 | C42 |
| Frattini, cosocle | C2×C82 | C22 |
| prime radical | C23 | #331 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C82 ▷ C42 ▷ C22 ▷ C1 | |
182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 32 points:
(2,7,8,11)(3,4)(5,13)(9,16,15,10)(12,14)(18,23,24,31)(19,27)(20,21)(25,28,26,32)(29,30);(1,2,4,9,13,16,3,11,6,8,14,15,5,10,12,7)(17,18)(19,28)(20,26)(21,25)(22,24)(23,29)(27,31)(30,32)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,e^2=fg,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72214503, #72214625, #72214696, … |
| maximal normal subgroup of | #72214503, #72214625, #72214696, … |
| maximal quotient of | #72214458, #72216557 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269);