#439574

order	512
nilpotency class	3
isoclinism	isoclinic class of #422715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1108 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=efg,c^2=ef,d^2=e,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, 346770);