#6368276

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368046
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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340115, #72340123, #72340133
maximal normal subgroup of	#72340115, #72340123, #72340133

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275472);