#368940

order	512
nilpotency class	3
isoclinism	isoclinic class of #196152
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

1130 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)(9,10,15,20,13,18,14,19)(11,21,24,34,23,16,28,31)(12,25,27,33,26,35,29,17)(22,36,39,32,37,40,38,30);(2,5)(6,7)(9,11,15,24,13,23,14,28)(10,16,20,31,18,21,19,34)(12,22,27,39,26,37,29,38)(17,32,25,40,33,30,35,36);(9,12,15,29,13,26,14,27)(10,17,19,25,18,33,20,35)(11,22,24,38,23,37,28,39)(16,30,34,36,21,32,31,40);(1,3,4,8)(2,6,5,7)(10,18)(16,21)(17,33)(19,20)(25,35)(30,32)(31,34)(36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gi,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, 276136);