#230431

order	512
nilpotency class	3
isoclinism	isoclinic class of #20158
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

575 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137627);