#6347289

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347252
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3830 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=i,e^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=di,f^a=fh,e^b=eh,f^b=fh,g^b=gi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281528, #72281704
maximal normal subgroup of	#72281528, #72281704

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254485);