#231471

order	512
nilpotency class	3
isoclinism	isoclinic class of #3288
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2366
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

679 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=f,d^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=de,e^b=eh,g^b=gh,d^c=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138667);