Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481373 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1151 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,19,20,36)(12,15)(13,25)(14,28)(16,21,30,38)(17,27)(18,24)(22,33,29,37)(23,35)(26,34,40,32)(31,39);(1,3,4,8)(2,5,6,7)(10,15)(11,20)(17,31)(18,28)(21,38)(22,29)(25,35)(26,40);(3,8)(5,7)(9,11)(10,16)(12,20)(13,26)(14,29)(15,30)(17,33)(18,34)(19,35)(21,39)(22,24)(23,40)(25,36)(27,38)(28,32)(31,37);(2,6)(5,7)(10,15)(16,30)(17,31)(18,28)(19,36)(25,35)(32,34)(33,37)Polycyclic presentation:
2^9|a^2=g,b^2=ei,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,f^c=fhi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72313749, #72313754, #72313784, … |
| maximal normal subgroup of | #72313749, #72313754, #72313784, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403560);