#6354225

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354207
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
prime radical	#60965	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1660 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)(3,11)(4,10)(5,7)(6,9)(8,12)(13,16)(14,15)(17,18)(19,25)(20,28)(21,27)(22,24)(23,31)(26,29)(30,32);(2,7)(8,15)(9,10)(11,16)(17,19,25,18,22,27,21,24)(20,26,31,23,29,28,30,32);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,11)(10,16)(17,20)(18,23)(19,26)(21,30)(22,29)(24,32)(25,31)(27,28);(1,4,5,6)(2,9,7,10)(3,12,13,14)(8,11,15,16)(18,24)(19,27)(23,32)(26,28);(20,29)(23,32)(26,28)(30,31)

Polycyclic presentation:

2^9|b^2=fi,d^2=hi,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^c=eh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306694, #72306696, #72306728
maximal normal subgroup of	#72306694, #72306696, #72306728

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261421);