#436856

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
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

790 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(2,5)(6,8)(9,10,13,15)(11,14,12,16)(18,23)(19,27)(21,30)(24,32)(25,29)(26,31);(1,2,4,5)(3,6,7,8)(9,11,13,12)(10,14,15,16)(17,18,22,26)(19,28,29,20)(21,31,30,23)(24,25,32,27);(1,3,4,7)(2,6,5,8)(10,15)(14,16)(17,19,21,27,22,29,30,25)(18,24,23,28,26,32,31,20);(2,5)(6,8)(9,12,13,11)(10,16,15,14)(17,20,30,32,22,28,21,24)(18,25,31,29,26,27,23,19)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 344052);