#6364216

order	512
nilpotency class	3
isoclinism	isoclinic class of #6363930
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

1537 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306948, #72306977, #72307025
maximal normal subgroup of	#72306948, #72306977, #72307025

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6271412);