Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20158 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | D4×C22⋊C4 |
| prime radical | #61436 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
867 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,7,6,3,10,9)(5,8,12,16,15,11,14,13)(17,18,22,25)(19,26,29,30)(20,28,31,23)(21,27,32,24);(1,3)(2,6)(4,9)(5,13)(7,10)(8,12)(11,14)(15,16)(17,19)(18,20)(21,28)(22,29)(23,32)(24,30)(25,31)(26,27);(1,4,6,10)(2,7,3,9)(5,12,15,14)(8,16,11,13)(17,20,22,31)(18,19,25,29)(21,26,32,30)(23,27,28,24);(1,5,6,15)(2,8,3,11)(4,12,10,14)(7,16,9,13)(17,20,22,31)(18,23,25,28)(19,27,29,24)(21,30,32,26)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ci,d^b=de,e^b=eh,f^c=fi,e^d=eh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72287015, #72287197, #72287201, … |
| maximal normal subgroup of | #72287015, #72287197, #72287201, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 128325);