#164468

order	512
nilpotency class	3
isoclinism	isoclinic class of #164187
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#550	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

956 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=gi,d^2=gi,e^2=h,f^2=i,b^a=be,c^a=cg,e^a=eg,e^b=eh,f^b=fgh,d^c=dgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 71664);