#431460

order	512
nilpotency class	3
isoclinism	isoclinic class of #418484
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1064 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)(3,7)(4,6)(5,8)(9,10,16,22,15,21,14,20)(11,23,13,17,26,19,25,34)(12,27,32,37,31,18,30,38)(24,36,29,35,28,39,40,33);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,28)(13,14)(15,26)(16,25)(18,35)(19,20)(21,34)(22,23)(24,31)(27,33)(29,32)(30,40)(36,38)(37,39);(2,6)(5,7)(9,12,16,32,15,31,14,30)(10,18,22,38,21,27,20,37)(11,24,13,29,26,28,25,40)(17,33,19,36,34,35,23,39);(3,8)(5,7)(9,13,16,26,15,25,14,11)(10,19,22,34,21,23,20,17)(12,29,32,28,31,40,30,24)(18,36,38,35,27,39,37,33)

Polycyclic presentation:

2^9|a^2=ef,c^2=ef,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, 338656);