#233595

order	512
nilpotency class	3
isoclinism	isoclinic class of #232771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1311 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fh,e^c=ei,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303252, #72303308, #72303341, …
maximal normal subgroup of	#72303252, #72303308, #72303341, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140791);