#237281

order	512
nilpotency class	3
isoclinism	isoclinic class of #237257
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	#60946	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1233 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)(3,12)(4,9)(5,10)(6,15)(7,11)(8,13)(14,16)(18,23)(19,27)(20,21)(25,26)(28,31)(29,30);(1,3)(2,8)(4,7)(5,13)(6,14)(9,10)(11,16)(12,15)(17,18)(19,28)(20,25)(21,23)(22,26)(24,27)(29,32)(30,31);(1,4,7,3)(2,9,11,12)(5,13,6,14)(8,15,16,10)(17,19,22,30)(18,24,26,32)(20,27,21,29)(23,28,25,31);(1,5,7,6)(2,10,11,15)(3,13,4,14)(8,9,16,12)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144477);