#368984

order	512
nilpotency class	3
isoclinism	isoclinic class of #196143
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1128 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)(3,7)(4,5)(6,8)(9,10,15,19,13,18,14,29)(11,20,24,37,22,16,23,32)(12,25,28,34,27,17,30,35)(21,36,26,33,38,31,39,40);(2,5)(6,7)(9,11,15,24,13,22,14,23)(10,16,19,32,18,20,29,37)(12,26,28,38,27,39,30,21)(17,33,35,31,25,40,34,36);(9,12,14,28,13,27,15,30)(10,17,29,35,18,25,19,34)(11,21,24,39,22,38,23,26)(16,31,32,33,20,36,37,40);(1,3,4,8)(2,6,5,7)(10,18)(16,20)(17,25)(19,29)(31,36)(32,37)(33,40)(34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276180);