Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419461 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1818 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)(6,11,14,12)(13,15,20,19);(5,6,8,12,10,14,9,11)(7,15,16,13,17,19,18,20);(6,12)(8,9)(11,14)(13,19)(15,20)(16,18);(2,4)(5,7)(6,13,14,20)(8,16)(9,18)(10,17)(11,15,12,19)Polycyclic presentation:
2^9|a^2=f,b^2=ef,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72219643, #72219665, #72219673, … |
| maximal normal subgroup of | #72219643, #72219665, #72219673, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337189);