#438560

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

793 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=ef,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,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, 345756);