#478190

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
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

682 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=ef,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^b=def,g^b=gi,d^c=de,g^c=gi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385386);