#164368

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×D₄²	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1133 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)(9,10,11,14,12,13,15,16)(17,18)(19,25,27,22)(20,28)(21,24)(23,29)(26,30,32,31);(1,2)(3,7)(4,6)(5,8)(10,13)(11,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32);(10,13)(14,16)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32);(20,29)(23,28)(26,32)(30,31)

Polycyclic presentation:

2^9|a^2=f,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, 71564);