#452122

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

631 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ef,c^2=e,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,h^b=hi,d^c=di

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, 359318);