Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624802 |
| 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×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1388 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)(3,7)(4,6)(5,8)(9,10)(11,20)(12,17)(13,27)(14,29)(15,19)(16,23)(18,21)(22,33)(24,39)(25,35)(26,32)(28,40)(30,37)(31,38)(34,36);(1,3)(2,5)(4,8)(6,7);(9,11,21,36,14,24,28,38,15,25,26,37,12,22,13,23)(10,16,27,33,17,30,32,35,19,31,40,39,29,34,18,20);(3,8)(5,7);(2,6)(5,7)(9,12,15,14)(10,17,19,29)(11,22,25,24)(13,26,28,21)(16,30,31,34)(18,27,32,40)(20,33,35,39)(23,37,38,36)Polycyclic presentation:
2^9|c^2=gh,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df104 conjugacy classes — view the full character table.
| maximal subgroup of | #72287855, #72314396, #72315808, … |
| maximal normal subgroup of | #72287855, #72314396, #72315808, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532005);