#164799

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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

694 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)(3,10,13,7)(4,14)(5,8)(6,9)(11,15,12,16)(17,18)(19,27,29,32)(20,25)(21,26)(22,24)(23,28,31,30)(33,34,37,41,38,42,36,40)(35,39,43,46,44,47,45,48);(1,3,6,13)(2,7,9,10)(4,11,5,12)(8,15,14,16)(17,19,22,29)(18,23,24,31)(20,28,21,30)(25,27,26,32);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(18,24)(23,31)(25,26)(27,32)(33,35,38,44)(34,39,42,47)(36,45,37,43)(40,48,41,46);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(17,20,22,21)(18,25,24,26)(19,28,29,30)(23,27,31,32)(33,36,38,37)(34,40,42,41)(35,43,44,45)(39,46,47,48)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=gi,d^2=ghi,e^2=h,f^2=i,b^a=be,c^a=cgh,e^a=eg,e^b=eh,f^b=fgh,d^c=di

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, 71995);