#431342

order	512
nilpotency class	3
isoclinism	isoclinic class of #418518
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

861 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=f,d^2=ef,e^2=i,g^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^b=gi,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, 338538);