Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476609 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| 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 | |
1152 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)(5,6,10,13,8,12,9,18)(7,14,17,20,16,11,15,19);(5,7,8,16)(6,11)(9,15,10,17)(12,14)(13,19)(18,20);(7,15)(9,10)(11,19)(13,18)(14,20)(16,17);(2,4)(6,12)(7,16)(9,10)(11,14)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,g^c=gi,g^d=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72318806, #72318947, #72318969, … |
| maximal normal subgroup of | #72318806, #72318947, #72318969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385621);