#230428

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2361
prime radical	#3035	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

577 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137624);