#244124

order	512
nilpotency class	3
isoclinism	isoclinic class of #232766
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2397
prime radical	#3100	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

623 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,d^2=ef,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,e^c=ei,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, 151320);