#164753

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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

756 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)(33,34,36,40,37,38,44,45)(35,41,43,48,42,39,46,47);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19,22,29)(18,23,24,31)(20,28,21,30)(25,27,26,32)(34,38)(36,44)(39,41)(43,46);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(18,24)(23,31)(25,26)(27,32)(33,35,37,42)(34,39,38,41)(36,43,44,46)(40,47,45,48);(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)(35,42)(39,41)(43,46)(47,48)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=g,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, 71949);