#379518

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

947 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=eg,d^2=ehi,e^2=i,f^2=i,g^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	#72226669, #72226672, #72226673
maximal normal subgroup of	#72226669, #72226672, #72226673

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286714);