#439433

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

826 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,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,d^b=dei,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, 346629);