#435304

order	512
nilpotency class	3
isoclinism	isoclinic class of #419843
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

669 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ef,c^2=ef,d^2=ei,g^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,g^b=gi,h^b=hi,g^c=gi

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, 342500);