#6349274

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349215
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2965 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)(11,16)(12,23)(13,19)(14,26)(15,18)(17,24)(20,35)(21,31)(22,30)(25,34)(27,32)(28,33)(29,38)(36,40)(37,39);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,20)(13,21)(14,27)(15,22)(17,29)(18,30)(19,31)(23,39)(24,36)(25,37)(26,32)(28,38)(33,40)(34,35);(9,12,15,25)(10,17,18,33)(11,20,22,37)(13,24,14,28)(16,29,30,40)(19,34,26,23)(21,36,27,38)(31,35,32,39);(1,3,5,4)(2,6,7,8)(10,18)(16,30)(17,33)(19,26)(23,34)(29,40)(31,32)(35,39);(2,7)(6,8)(9,13,15,14)(10,19,18,26)(11,21,22,27)(12,24,25,28)(16,31,30,32)(17,34,33,23)(20,36,37,38)(29,35,40,39)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306958, #72306975, #72307068
maximal normal subgroup of	#72306958, #72306975, #72307068

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256470);