#166989

order	512
nilpotency class	3
isoclinism	isoclinic class of #166289
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

691 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)(4,8)(5,14)(6,9)(7,11)(12,16)(13,15)(17,18,22,21)(19,23,27,26)(20,24,29,28)(25,31,32,30)(33,34)(35,42)(36,40)(37,45)(38,41)(39,43)(44,47)(46,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,16)(10,14)(17,19,22,27)(18,23,21,26)(20,25,29,32)(24,30,28,31);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,16,10)(17,20,22,29)(18,24,21,28)(19,25,27,32)(23,30,26,31)(33,35)(34,39)(36,43)(37,46)(38,44)(40,47)(41,48)(42,45);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,10,16,15)(17,21,22,18)(19,26,27,23)(20,28,29,24)(25,31,32,30)(33,36,38,37)(34,40,41,45)(35,43,44,46)(39,47,48,42)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=ghi,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 74185);