Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394098 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4407 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,12)(8,15)(9,13)(10,14)(11,16)(17,20)(18,19);(5,7)(6,11)(8,18)(9,16)(10,17)(12,15)(13,20)(14,19);(2,4)(5,7)(6,12)(8,18)(9,16)(10,17)(11,13)(14,20)(15,19);(5,7)(6,11)(8,16)(9,18)(10,17)(12,13)(14,19)(15,20)Polycyclic presentation:
2^9|e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72219645, #72219660, #72219662, … |
| maximal normal subgroup of | #72219645, #72219660, #72219662, … |
| maximal quotient of | #72226226, #72231222, #72231590, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301294);