#410764

order	512
nilpotency class	3
isoclinism	isoclinic class of #394465
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

637 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)(3,5)(4,6)(7,8)(9,10,14,20,15,19,30,21)(11,22,24,12,25,16,17,27)(13,18,28,34,29,33,40,35)(23,36,37,26,38,31,32,39);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,20)(13,23)(14,17)(15,25)(18,31)(19,22)(21,27)(24,30)(26,34)(28,32)(29,38)(33,36)(35,39)(37,40);(9,12,14,22,15,27,30,16)(10,17,21,11,19,24,20,25)(13,26,28,36,29,39,40,31)(18,32,35,23,33,37,34,38);(2,6)(5,8)(9,13,14,28,15,29,30,40)(10,18,20,34,19,33,21,35)(11,23,17,32,25,38,24,37)(12,26,22,36,27,39,16,31)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317960);