Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18682 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#379 |
| prime radical | #62762 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,6,4,5,9)(3,8,12,15,13,11,14,16)(17,18)(19,25,27,32)(20,24)(21,23)(22,26,30,28)(29,31);(1,3)(2,8)(4,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,19)(18,22)(20,28)(21,27)(23,30)(24,32)(25,31)(26,29);(1,4)(2,6)(3,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,18)(19,22)(20,24)(21,23)(25,26)(27,30)(28,32)(29,31);(1,5,6,7)(2,9,4,10)(3,12,13,14)(8,15,11,16)(18,23)(22,30)(24,31)(25,32)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=ghi,d^c=di,e^c=eh,f^c=fh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72209982, C2×#186596, #72272564, … |
| maximal normal subgroup of | #72209982, C2×#186596, #72272564, … |
| maximal quotient of | C2×#186596, #72238829, #72238884, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 93792);