#6354297

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

Polycyclic presentation:

2^9|b^2=fi,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	#72306584, #72306613, #72306716, …
maximal normal subgroup of	#72306584, #72306613, #72306716, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261493);