Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #61436 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #2965 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1583 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,12)(3,10,5,14)(4,11,6,9)(8,15,16,13)(17,18,22,19,21,23,29,25)(20,24,28,26,27,30,32,31);(1,3,7,5)(2,8,12,16)(4,13,6,15)(9,10,11,14)(18,23)(19,25)(24,30)(26,31);(1,4,7,6)(2,9,12,11)(3,13,5,15)(8,10,16,14)(17,19,22,23,21,25,29,18)(20,26,28,30,27,31,32,24);(1,5,7,3)(2,10,12,14)(4,13,6,15)(8,11,16,9)(17,20)(18,24)(19,26)(21,27)(22,28)(23,30)(25,31)(29,32);(1,4,7,6)(2,11,12,9)(3,13,5,15)(8,14,16,10)(20,27)(24,30)(26,31)(28,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=chi,d^a=dh,f^a=fh,c^b=ci,f^b=fh,d^c=dh,e^d=ehi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72286949, #72286969, #72287018, … |
| maximal normal subgroup of | #72286949, #72286969, #72287018, … |
| maximal quotient of | #72343703, #72343763 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253221);