#164812

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

703 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)(19,26)(20,23)(21,25)(22,30)(24,28)(27,31)(29,32)(33,34,37,42,38,41,36,40)(35,43,44,48,45,39,46,47);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19)(18,24)(20,27)(21,28)(22,31)(23,29)(25,32)(26,30)(33,35)(34,39)(36,44)(37,46)(38,45)(40,47)(41,43)(42,48);(2,7)(6,14)(8,15)(9,16)(17,20,23,18)(19,27,29,24)(21,25,22,30)(26,28,32,31)(33,36,38,37)(34,40,41,42)(35,44,45,46)(39,47,43,48);(17,21,23,22)(18,25,20,30)(19,28,29,31)(24,32,27,26)

Polycyclic presentation:

2^9|a^2=f,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, 72008);