#453477

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

644 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=e,c^2=g,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,h^c=hi,e^d=ei,f^d=fi,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, 360673);