#230419

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

592 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:

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

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=dhi,f^a=fi,d^b=de,e^b=eh,g^b=gh,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, 137615);