Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422233 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| 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 | |
1330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,10)(7,14)(8,17)(11,13)(12,15)(18,20);(1,2,3,4)(5,6,9,13)(7,15,14,18)(8,11,17,10)(12,19,20,16);(2,4)(6,11,13,10)(12,18,20,15);(5,7,8,16,9,14,17,19)(6,12,10,15,13,20,11,18)Polycyclic presentation:
2^9|b^2=eg,c^2=f,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72320044, #72320049, #72320106, … |
| maximal normal subgroup of | #72320044, #72320049, #72320106, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 345530);