Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×#509 |
| 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 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1421 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)(11,16)(12,22)(13,25)(14,29)(15,18)(17,23)(19,27)(20,36)(21,34)(24,39)(26,31)(28,38)(30,35)(32,37)(33,40);(1,2,5,6)(3,7,4,8)(9,11,15,14)(10,16,18,29)(12,20,24,30)(13,21,28,27)(17,31,32,33)(19,25,34,38)(22,36,39,35)(23,26,37,40);(1,3,5,4)(2,7,6,8)(9,12,15,24)(10,17,18,32)(11,20,14,30)(13,26,28,40)(16,31,29,33)(19,35,34,36)(21,37,27,23)(22,38,39,25);(1,4,5,3)(2,7,6,8);(10,18)(16,29)(17,32)(19,34)(22,39)(25,38)(31,33)(35,36)Polycyclic presentation:
2^9|b^2=fi,c^2=fi,d^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,d^b=df,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72309518, #72309533, #72309541, … |
| maximal normal subgroup of | #72309518, #72309533, #72309541, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531160);