#230427

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	#19157	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

606 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=eh,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, 137623);