#466369

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
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

764 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,14,12,16)(17,18,21,27)(19,24,31,40)(20,34,35,25)(22,28,23,38)(26,37,43,36)(29,44,45,39)(30,46,47,42)(32,41,33,48);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19,23,33,21,31,22,32)(18,24,38,48,27,40,28,41)(20,30,37,29,35,47,36,45)(25,42,26,39,34,46,43,44);(1,3,4,7)(2,6,5,8)(10,15)(14,16)(17,20)(18,25)(19,29)(21,35)(22,37)(23,36)(24,39)(26,28)(27,34)(30,33)(31,45)(32,47)(38,43)(40,44)(41,46)(42,48);(3,7)(6,8)(9,12,13,11)(10,16,15,14)(17,20)(18,26)(19,30)(21,35)(22,36)(23,37)(24,39)(25,28)(27,43)(29,33)(31,47)(32,45)(34,38)(40,44)(41,42)(46,48)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=e,d^2=fi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,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, 373565);