#246348

order	512
nilpotency class	3
isoclinism	isoclinic class of #246319
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1888 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gh,c^b=cg,e^b=eh,f^b=fhi,e^c=ei,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255902, #72256141, #72256168
maximal normal subgroup of	#72255902, #72256141, #72256168

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153544);