#477716

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
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

876 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=fi,c^2=ef,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=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, 384912);