#466629

order	512
nilpotency class	3
isoclinism	isoclinic class of #457102
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

931 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)(11,20)(12,15)(13,27)(14,18)(16,22)(17,29)(19,25)(21,32)(23,33)(24,31)(26,38)(28,30)(34,35)(36,37)(39,40);(2,5)(6,8)(9,11,15,23,14,22,25,31)(10,16,12,24,18,20,19,33)(13,21,30,36,29,35,38,40)(17,32,26,37,27,34,28,39);(1,3)(2,6)(4,7)(5,8)(9,12,25,18,14,19,15,10)(11,16,31,24,22,20,23,33)(13,28,38,17,29,26,30,27)(21,34,40,39,35,32,36,37);(3,7)(6,8)(9,13,15,30,14,29,25,38)(10,17,19,28,18,27,12,26)(11,21,23,36,22,35,31,40)(16,32,33,39,20,34,24,37)

Polycyclic presentation:

2^9|b^2=g,c^2=fi,d^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^c=de,g^c=gi,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, 373825);