#244432

order	512
nilpotency class	3
isoclinism	isoclinic class of #232749
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	C₂²×D₄	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

579 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:

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

Polycyclic presentation:

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

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, 151628);