#428242

order	512
nilpotency class	3
isoclinism	isoclinic class of #418484
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1015 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,c^2=e,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^c=gi,h^d=hi

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, 335438);