Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220548 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #61410 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
835 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,3,11,9)(4,8,15,12,16,13,14,7)(17,18,24,29)(19,30,25,32)(20,31,21,26)(22,28,23,27);(1,3)(2,6)(4,7)(5,9)(8,15)(10,11)(12,16)(13,14)(17,19,24,25)(18,20,29,21)(22,31,23,26)(27,32,28,30);(1,4)(2,7)(3,12)(5,14)(6,16)(8,9)(10,13)(11,15)(17,20,24,21)(18,25,29,19)(22,30,23,32)(26,27,31,28);(1,4)(2,8)(3,13)(5,15)(6,16)(7,9)(10,12)(11,14)(17,21,24,20)(18,26,29,31)(19,28,25,27)(22,30,23,32)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=i,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=de,e^b=eh,f^c=fh,g^c=ghi,e^d=eh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 127896);