#379520

order	512
nilpotency class	3
isoclinism	isoclinic class of #219491
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

902 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)(3,7)(4,8)(5,6)(9,10,14,17,15,20,13,19)(11,21,24,30,25,32,22,16)(12,26,28,34,29,33,27,18)(23,37,38,40,39,31,36,35);(2,6)(7,8)(9,11,13,22,15,25,14,24)(10,16,17,21,20,30,19,32)(12,23,27,36,29,39,28,38)(18,31,26,35,34,37,33,40);(1,3,5,4)(2,7,6,8)(10,17)(11,22)(13,14)(16,30)(18,26)(19,20)(23,36)(24,25)(27,28)(31,37)(33,34)(38,39);(1,4,5,3)(2,7,6,8)(9,12)(10,18,20,34)(11,23)(13,27)(14,28)(15,29)(16,31,30,37)(17,26,19,33)(21,35,32,40)(22,36)(24,38)(25,39)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226668, #72226673, #72226674
maximal normal subgroup of	#72226668, #72226673, #72226674

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286716);