#458444

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

793 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=efi,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,g^c=gi,h^d=hi

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, 365640);