#6365441

order	512
nilpotency class	3
isoclinism	isoclinic class of #6365322
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

1610 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=hi,e^2=h,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=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283494
maximal normal subgroup of	#72283494

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6272637);