#440649

order	512
nilpotency class	3
isoclinism	isoclinic class of #422715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

965 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=efg,c^2=e,d^2=efi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi,h^d=hi

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, 347845);