#230410

order	512
nilpotency class	3
isoclinism	isoclinic class of #19755
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

596 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,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,f^c=fi

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, 137606);