#6354284

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354266
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
prime radical	#60962	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1661 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=fi,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,d^b=dh,e^b=eh,e^c=eh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306686, #72306693, #72306702
maximal normal subgroup of	#72306686, #72306693, #72306702

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261480);