#234252

order	512
nilpotency class	3
isoclinism	isoclinic class of #234235
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	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,6,10)(3,11,14,15)(4,9,5,8)(7,13,16,12)(17,18,22,26)(19,27,29,42)(20,24,32,41)(21,33,36,25)(23,34,38,30)(28,44,45,48)(31,43,35,40)(37,46,39,47);(1,3,6,14)(2,7,10,16)(4,13,5,12)(8,15,9,11)(17,19,22,29)(18,23,26,38)(20,28,32,45)(21,34,36,30)(24,39,41,37)(25,42,33,27)(31,47,35,46)(40,48,43,44);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(17,20,22,32)(18,24,26,41)(19,28,29,45)(21,35,36,31)(23,37,38,39)(25,43,33,40)(27,44,42,48)(30,47,34,46);(19,29)(23,38)(27,42)(28,45)(30,34)(37,39)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,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=fi,e^c=eh,f^c=fi

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, 141448);