#439377

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

887 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)(6,7)(9,10,13,15)(11,14,12,16)(17,18)(19,27)(20,22)(21,30)(23,26)(24,31)(25,28)(29,32);(1,2,4,5)(3,7,8,6)(9,11,13,12)(10,14,15,16)(17,19,23,29,22,28,24,30)(18,25,26,21,20,27,31,32);(9,12,13,11)(10,14,15,16)(17,20,22,18)(19,21,28,32)(23,26,24,31)(25,30,27,29);(1,3,4,8)(2,6,5,7)(17,21,22,32)(18,19,20,28)(23,27,24,25)(26,29,31,30)

Polycyclic presentation:

2^9|a^2=fi,b^2=efg,c^2=f,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, 346573);