#446065

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

942 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)(4,7)(5,8)(9,10,15,20,16,12,14,19)(11,21,25,17,26,22,24,27)(13,18,30,35,31,28,32,36)(23,34,29,40,38,37,39,33);(1,3,5,4)(2,6,8,7)(9,11)(10,17)(12,27)(13,29)(14,25)(15,24)(16,26)(18,34)(19,22)(20,21)(23,30)(28,37)(31,39)(32,38)(33,35)(36,40);(1,3,5,4)(2,7,8,6)(9,12,15,19,16,10,14,20)(11,22,25,27,26,21,24,17)(13,28,30,36,31,18,32,35)(23,34,29,40,38,37,39,33);(1,4,5,3)(2,7,8,6)(9,13)(10,18)(11,23)(12,28)(14,32)(15,30)(16,31)(17,33)(19,36)(20,35)(21,37)(22,34)(24,39)(25,29)(26,38)(27,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 353261);