#452690

order	512
nilpotency class	3
isoclinism	isoclinic class of #418488
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

671 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)(11,16,22,32)(12,24,26,17)(13,19)(14,20)(15,18)(21,37)(23,33,29,31)(25,40)(27,36,28,35)(30,38)(34,39);(1,2)(3,7)(4,8)(5,6)(9,11,14,23,13,22,15,29)(10,16,20,33,19,32,18,31)(12,25,27,38,26,39,28,21)(17,34,35,37,24,40,36,30);(1,3,5,4)(2,7,6,8)(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)(31,34,33,40);(1,4,5,3)(2,7,6,8)(10,18,19,20)(11,22)(16,31,32,33)(17,35,24,36)(21,38)(23,29)(25,39)(30,34,37,40)

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,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,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, 359886);