#169534

order	512
nilpotency class	3
isoclinism	isoclinic class of #168909
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	#2682	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

693 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=hi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 76730);