#244144

order	512
nilpotency class	3
isoclinism	isoclinic class of #232771
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

618 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151340);