#444467

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1204 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 351663);