#466925

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

706 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=gi,c^2=f,d^2=f,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, 374121);