#444475

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1324 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=fgi,d^2=fi,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 351671);