#6369176

order	512
nilpotency class	3
isoclinism	isoclinic class of #6369150
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1599 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306663, #72306690, #72306716
maximal normal subgroup of	#72306663, #72306690, #72306716

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276372);