#164872

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	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

692 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18)(19,25)(20,29)(21,26)(22,31)(23,27)(24,28)(30,32)(33,34,37,42,38,41,36,40)(35,43,44,48,45,39,46,47);(1,3,5,11)(2,6,7,14)(4,12,13,10)(8,16,15,9)(17,19,23,20)(18,24,27,32)(21,28,22,30)(25,26,29,31)(33,35)(34,39)(36,44)(37,46)(38,45)(40,47)(41,43)(42,48);(2,7)(6,14)(8,15)(9,16)(17,20,23,19)(18,25,27,29)(21,28,22,30)(24,26,32,31)(33,36,38,37)(34,40,41,42)(35,44,45,46)(39,47,43,48);(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=g,d^2=gi,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, 72068);