#237131

order	512
nilpotency class	3
isoclinism	isoclinic class of #237115
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1208 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,d^b=dhi,e^b=eh,f^b=fhi,g^b=gh,e^c=ei,f^c=fhi,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144327);