#6369171

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
prime radical	#62328	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1591 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)(3,9)(4,13)(5,7)(6,11)(8,14)(10,16)(12,15)(17,18)(19,23)(20,25)(21,24)(22,26)(27,29)(28,32)(30,31);(1,3,5,12)(2,6,7,16)(4,10,14,11)(8,15,13,9)(17,19,22,28)(18,23,26,32)(20,29,30,21)(24,25,27,31);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,16)(13,14)(18,24)(19,28)(21,29)(23,31)(25,32)(26,27);(3,10)(4,14)(6,15)(8,13)(9,16)(11,12)(17,20)(18,25)(19,29)(21,28)(22,30)(23,27)(24,32)(26,31);(2,7)(6,16)(8,13)(9,15)(18,26)(19,28)(20,30)(24,27)

Polycyclic presentation:

2^9|b^2=hi,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	#72306662, #72306663, #72306724
maximal normal subgroup of	#72306662, #72306663, #72306724

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276367);