#240626

order	512
nilpotency class	3
isoclinism	isoclinic class of #240625
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

956 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,5)(4,7)(6,8)(9,10,14,18)(11,19,21,34)(12,23,25,16)(13,27,29,17)(15,28,31,22)(20,36,24,35)(26,33,40,38)(30,37,32,39);(3,8)(5,6)(9,11)(10,15)(12,24)(13,28)(14,21)(16,32)(17,34)(18,31)(19,27)(20,25)(22,29)(23,30)(26,37)(33,35)(36,38)(39,40);(2,5)(3,8)(6,7)(9,12,13,26,14,25,29,40)(10,16,17,33,18,23,27,38)(11,20,22,37,21,24,28,39)(15,30,19,35,31,32,34,36);(11,21)(12,25)(15,31)(16,23)(19,34)(22,28)(26,40)(33,38)

Polycyclic presentation:

2^9|a^2=h,c^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,d^c=dh,f^c=fhi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 147822);