#437217

order	512
nilpotency class	3
isoclinism	isoclinic class of #419843
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

789 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)(3,7,8,6)(9,10,13,20,15,19,14,18)(11,21,23,32,25,16,24,33)(12,26,28,36,29,17,30,35)(22,31,27,34,38,37,39,40);(1,3)(2,6)(4,7)(5,8)(9,11,15,25)(10,16,19,21)(12,27,29,39)(13,24,14,23)(17,34,26,40)(18,33,20,32)(22,30,38,28)(31,36,37,35);(1,4,5,2)(3,7,8,6)(9,12)(10,17)(11,22)(13,30)(14,28)(15,29)(16,31)(18,35)(19,26)(20,36)(21,37)(23,39)(24,27)(25,38)(32,34)(33,40);(3,8)(6,7)(10,18,19,20)(16,32,21,33)(17,35,26,36)(31,34,37,40)

Polycyclic presentation:

2^9|a^2=efg,b^2=i,c^2=ef,d^2=f,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, 344413);