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 | #62761 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,9,5,4,13,8)(3,7,12,16,11,10,14,15)(17,18)(19,25,27,32)(20,24)(21,23)(22,26,30,28)(29,31);(1,3,5,11)(2,7,4,10)(6,14,13,12)(8,16,9,15)(17,19)(18,22)(20,28)(21,27)(23,30)(24,32)(25,31)(26,29);(1,4)(2,5)(3,10)(6,8)(7,11)(9,13)(12,15)(14,16)(17,18)(19,22)(20,24)(21,23)(25,26)(27,30)(28,32)(29,31);(18,23)(22,30)(24,31)(25,32)Polycyclic presentation:
2^9|a^2=g,b^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×#186595 |
| maximal normal subgroup of | #72209982, C2×#186595 |
| maximal quotient of | C2×#186595, #72238827, #72238886, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 93791);