#468992

order	512
nilpotency class	3
isoclinism	isoclinic class of #455826
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
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

694 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 376188);