Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623613 |
| 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 ◁ #532 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,13,19)(11,22,23,16)(12,18,26,37)(14,17,28,35)(15,30,24,21)(20,38,36,29)(25,31,39,33)(27,34,40,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,15)(13,23)(14,25)(17,33)(18,21)(19,22)(20,32)(24,26)(27,29)(28,39)(30,37)(31,35)(34,36)(38,40);(10,17)(11,23)(14,29)(15,24)(16,31)(18,34)(19,35)(20,21)(22,33)(25,40)(27,39)(28,38)(30,36)(32,37);(3,8)(5,7)(9,12)(10,18)(11,24)(13,26)(14,27)(15,23)(16,30)(17,34)(19,37)(20,33)(21,22)(25,38)(28,40)(29,39)(31,36)(32,35);(10,19)(11,23)(12,26)(17,35)(20,36)(21,30)(25,39)(27,40)Polycyclic presentation:
2^9|a^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei,g^c=gh,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72216687, C2×#7623777, #72320147, … |
| maximal normal subgroup of | #72216687, C2×#7623777, #72320147, … |
| maximal quotient of | C2×#7623777, #72319515, #72319537, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530973);