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 ◁ C4×#41 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10,14,17)(11,15,22,28)(12,16,25,31)(13,26,24,19)(18,33,32,23)(20,27,36,38)(21,37,35,30)(29,40,39,34);(1,2)(3,6)(4,5)(7,8)(9,11,14,22)(10,15,17,28)(12,20,25,36)(13,21,24,35)(16,27,31,38)(18,29,32,39)(19,30,26,37)(23,34,33,40);(10,16)(12,23)(13,24)(15,27)(17,31)(18,19)(20,34)(21,35)(25,33)(26,32)(28,38)(29,30)(36,40)(37,39);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(10,17)(15,28)(16,31)(18,32)(19,26)(27,38)(29,39)(30,37)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^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 | #72321472, #72321618, #72321751, … |
| maximal normal subgroup of | #72321472, #72321618, #72321751, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531274);