Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7092 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #331 |
| commutator | C4×C8 | C2×C8 |
| Frattini, cosocle | C2×C82 | C22 |
| socle | C22 | #1270 |
| prime radical | C23 | #329 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C82 ▷ C2×C42 ▷ C22 ▷ C1 | |
179 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,10,6,11,3,13,7,12,14,8,5,9,15,16)(18,22,23,30,25,31,24,26);(1,3,7,15)(2,8,12,10)(4,5,14,6)(9,16,11,13)(17,18)(19,26)(20,23)(21,25)(22,28)(24,27)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=d,b^2=h,c^2=fi,d^2=g,e^2=h,f^2=i,g^2=h,b^a=bc,c^a=ce,e^a=eh,f^a=fh,c^b=cf,d^b=defh,e^b=eh,f^b=fi,g^b=ghi86 conjugacy classes — view the full character table.
| maximal quotient of | #72210667, #72210671 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 61);