#179369

order	512
nilpotency class	3
isoclinism	isoclinic class of #179229
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	#53182	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

719 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)(3,8)(4,9)(5,7)(6,10)(11,14)(12,16)(13,15)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,31,32,28)(33,34)(35,43)(36,40)(37,41)(38,42)(39,45)(44,47)(46,48);(2,7)(4,5)(8,14)(9,10)(11,13)(15,16)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,32)(30,31)(33,35)(34,39)(36,44)(37,45)(38,46)(40,43)(41,48)(42,47);(1,3,6,12)(2,8,10,16)(4,13,5,11)(7,14,9,15)(17,20,21,27)(18,23,24,30)(19,25,26,32)(22,28,29,31)(33,35)(34,39)(36,45)(37,44)(38,46)(40,48)(41,43)(42,47);(1,4,6,5)(2,9,10,7)(3,11,12,13)(8,14,16,15)(33,36,38,37)(34,40,42,41)(35,44,46,45)(39,43,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 86565);