#6362727

order	512
nilpotency class	3
isoclinism	isoclinic class of #6362510
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

1612 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283267
maximal normal subgroup of	#72283267

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6269923);