#6368932

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368846
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

1556 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ehi,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	#72306613, #72306700, #72306709
maximal normal subgroup of	#72306613, #72306700, #72306709

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276128);