#368936

order	512
nilpotency class	3
isoclinism	isoclinic class of #196299
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	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1184 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)(11,21,24,33,23,16,30,32)(12,25,28,35,27,36,29,17)(22,34,39,37,38,40,26,31);(2,5)(6,7)(9,11,15,24,13,23,14,30)(10,16,20,32,18,21,19,33)(12,26,28,22,27,39,29,38)(17,34,25,37,35,40,36,31);(9,12)(10,17,18,35)(11,22,23,38)(13,27)(14,28)(15,29)(16,31)(19,25,20,36)(21,37)(24,26,30,39)(32,40)(33,34);(1,3,4,8)(2,6,5,7)(10,18)(16,21)(17,35)(19,20)(25,36)(31,37)(32,33)(34,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,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, 276132);