Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623604 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(7,8)(9,10,14,19)(11,15,20,28)(12,16,25,32)(13,26,24,18)(17,33,31,23)(21,27,34,38)(22,37,35,30)(29,40,39,36);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,27)(17,29)(18,30)(19,28)(23,36)(24,35)(25,34)(26,37)(31,39)(32,38)(33,40);(1,3,5,4)(2,7,6,8)(10,16)(12,23)(13,24)(15,27)(17,18)(19,32)(21,36)(22,35)(25,33)(26,31)(28,38)(29,30)(34,40)(37,39);(1,4,5,3)(2,7,6,8);(11,20)(15,28)(21,34)(22,35)(27,38)(29,39)(30,37)(36,40)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,d^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72322060 |
| maximal normal subgroup of | #72322060 |
| maximal quotient of | #72343826 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531162);