#457025

order	512
nilpotency class	3
isoclinism	isoclinic class of #456417
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
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

1098 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)(3,6)(4,5)(7,8)(9,10,14,19,15,13,16,29)(11,17,22,32,24,21,23,38)(12,25,27,26,28,33,30,18)(20,34,36,35,37,40,39,31);(2,5)(6,8)(9,11,14,22,15,24,16,23)(10,17,19,32,13,21,29,38)(12,20,27,36,28,37,30,39)(18,31,25,34,26,35,33,40);(3,7)(6,8)(9,12)(10,18)(11,20)(13,26)(14,30)(15,28)(16,27)(17,31)(19,33)(21,35)(22,39)(23,36)(24,37)(25,29)(32,40)(34,38);(1,3,4,7)(2,6,5,8)(9,13,15,10)(11,21,24,17)(12,26,28,18)(14,29,16,19)(20,35,37,31)(22,38,23,32)(25,27,33,30)(34,36,40,39)

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 364221);