Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417562 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2002 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1628 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)(8,17)(10,13)(11,14)(16,20);(5,6)(7,14)(8,10)(9,12)(11,15)(13,17)(16,19)(18,20);(6,10)(8,17)(11,18)(12,13)(14,19)(16,20);(2,4)(5,7)(6,11,12,14)(8,16)(9,15)(10,18,13,19)(17,20)Polycyclic presentation:
2^9|d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72219643, #72219657, #72219665, … |
| maximal normal subgroup of | #72219643, #72219657, #72219665, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334438);