#230429

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,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, 137625);