#164785

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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

740 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)(4,8)(5,7)(10,15,16,13)(12,14)(17,18)(19,26,28,31)(20,29)(21,24)(22,25)(23,27,32,30)(33,34,37,41,38,42,36,40)(35,39,43,46,44,47,45,48);(1,3,5,11)(2,6,7,9)(4,10,12,16)(8,13,14,15)(17,19,22,28)(18,23,25,32)(20,27,21,30)(24,31,29,26);(2,7)(6,9)(8,14)(13,15)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)(33,35)(34,39)(36,45)(37,43)(38,44)(40,48)(41,46)(42,47);(1,4,5,12)(2,8,7,14)(3,10,11,16)(6,13,9,15)(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=h,d^2=gi,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, 71981);