#230422

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,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, 137618);