#462009

order	512
nilpotency class	3
isoclinism	isoclinic class of #454986
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

651 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)(3,6)(4,8)(5,7)(9,10,14,20,13,19,15,18)(11,16,23,31,22,32,29,33)(12,24,27,36,26,17,28,35)(21,37,25,40,38,30,39,34);(1,3,5,4)(2,6,7,8)(9,11,13,22)(10,16,19,32)(12,25,26,39)(14,29,15,23)(17,34,24,40)(18,31,20,33)(21,28,38,27)(30,36,37,35);(1,4,5,3)(2,6,7,8)(9,12,13,26)(10,17,19,24)(11,21,22,38)(14,28,15,27)(16,30,32,37)(18,35,20,36)(23,39,29,25)(31,40,33,34);(10,18,19,20)(12,26)(16,31,32,33)(17,35,24,36)(21,38)(25,39)(27,28)(30,40,37,34)

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,g^b=gi,d^c=de,g^c=gi,h^c=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, 369205);