#6354135

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354110
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	#60951	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1620 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)(4,9)(5,8)(6,10)(7,12)(13,15)(14,16)(17,18,22,25)(19,21,24,31)(20,27,30,26)(23,29,32,28);(17,19,21,25,22,24,31,18)(20,26,29,32,30,27,28,23);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,15)(10,16)(17,20)(18,23)(19,26)(21,29)(22,30)(24,27)(25,32)(28,31);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,11,16,15)(17,21,22,31)(18,19,25,24)(20,28,30,29)(23,27,32,26);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,15,16,11)(17,21,22,31)(18,24,25,19)(20,29,30,28)(23,27,32,26)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306668, #72306673, #72306675, …
maximal normal subgroup of	#72306668, #72306673, #72306675, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261331);