#443512

order	512
nilpotency class	3
isoclinism	isoclinic class of #420420
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1102 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=g,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=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, 350708);