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 | #61430 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
871 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,5)(8,15)(9,13)(10,14)(11,16)(17,18,21,25,22,24,20,29)(19,26,28,23,27,32,30,31);(1,2,6,9)(3,7,5,13)(4,11,12,15)(8,14,16,10)(19,27)(23,31)(26,32)(28,30);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,19)(18,23)(20,30)(21,28)(22,27)(24,31)(25,32)(26,29);(1,4,6,12)(2,8,9,16)(3,10,5,14)(7,15,13,11)(18,24)(23,31)(25,29)(26,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=h,f^2=i,g^2=i,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 | #72287242, #72287279, #72287310, … |
| maximal normal subgroup of | #72287242, #72287279, #72287310, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 128285);