#6354152

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354139
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	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1633 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(18,23)(19,27)(20,21)(25,26)(28,29)(30,31);(17,18,21,23,22,26,20,25)(19,24,29,30,28,32,27,31);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19,22,28)(18,24,26,32)(20,27,21,29)(23,30,25,31);(2,7)(6,15)(8,12)(9,16)(17,20,22,21)(18,25,26,23)(19,27,28,29)(24,31,32,30);(19,28)(24,32)(27,29)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306666, #72306667, #72306683, …
maximal normal subgroup of	#72306666, #72306667, #72306683, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261348);