Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395173 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2404 |
| prime radical | #23443 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,20);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,17)(13,19)(16,20);(10,17)(18,19);(6,11,12,13)(10,18,17,19)Polycyclic presentation:
2^9|b^2=e,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,f^b=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216067, #72216224, #72219644, … |
| maximal normal subgroup of | #72216067, #72216224, #72219644, … |
| maximal quotient of | #72215966, #72224911 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311236);