#452964

order	512
nilpotency class	3
isoclinism	isoclinic class of #421163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

693 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)(6,8)(9,10,15,28,13,18,14,19)(11,20,21,32,23,16,24,30)(12,17,27,35,26,34,29,38)(22,33,37,39,25,31,36,40);(1,2,4,5)(3,6,7,8)(9,11,15,21,13,23,14,24)(10,16,28,30,18,20,19,32)(12,25,29,37,26,22,27,36)(17,33,38,40,34,31,35,39);(1,3,4,7)(2,6,5,8)(11,21)(14,15)(16,30)(19,28)(20,32)(22,36)(23,24)(25,37)(27,29)(31,39)(33,40)(35,38);(2,5)(6,8)(9,12,13,26)(10,17)(11,22,23,25)(14,27,15,29)(16,31)(18,34)(19,35)(20,33)(21,36,24,37)(28,38)(30,39)(32,40)

Polycyclic presentation:

2^9|a^2=fgh,b^2=eg,c^2=ei,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,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, 360160);