#6354294

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354266
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1652 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306568, #72306601, #72306716, …
maximal normal subgroup of	#72306568, #72306601, #72306716, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261490);