#164873

order	512
nilpotency class	3
isoclinism	isoclinic class of #164512
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	#2407	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

793 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)(4,7)(5,8)(10,15,16,13)(12,14)(17,18)(19,26,28,31)(20,29)(21,24)(22,25)(23,27,32,30)(33,34,37,42,38,41,36,40)(35,43,44,48,45,39,46,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,22,28)(18,23,25,32)(20,27,21,30)(24,31,29,26)(33,35)(34,39)(36,44)(37,46)(38,45)(40,47)(41,43)(42,48);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26);(4,12)(7,14)(10,16)(13,15)(33,36,38,37)(34,40,41,42)(35,44,45,46)(39,47,43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 72069);