#243840

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	C₂×#399
prime radical	C₂×#121	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

647 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ef,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, 151036);