#164970

order	512
nilpotency class	3
isoclinism	isoclinic class of #164187
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	#2414	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

777 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,9,11,6)(4,12)(5,8)(7,13)(10,14,16,15)(17,18)(19,26)(20,25)(21,29)(22,24)(23,27)(28,32)(30,31)(33,34,37,40,38,41,36,45)(35,42,43,47,44,39,46,48);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,19,22,28)(18,23,24,31)(20,27,21,30)(25,32,29,26)(33,35,38,44)(34,39,41,42)(36,43,37,46)(40,47,45,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(18,24)(23,31)(25,29)(26,32)(33,36,38,37)(34,40,41,45)(35,43,44,46)(39,47,42,48);(4,13)(7,12)(10,16)(14,15)(17,20,22,21)(18,25,24,29)(19,27,28,30)(23,32,31,26)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gh,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cg,e^a=eg,e^b=eh,f^b=fgh,d^c=dghi

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, 72166);