#6354165

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1921 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306665, #72306684, #72306719
maximal normal subgroup of	#72306665, #72306684, #72306719

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261361);