#173879

order	512
nilpotency class	3
isoclinism	isoclinic class of #173481
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

647 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)(19,26)(20,24)(21,29)(22,25)(23,27)(28,31)(30,32)(33,34,39,44)(35,45,47,41)(36,48,46,40)(37,43,38,42);(1,3,6,13)(2,7,9,16)(4,11,5,12)(8,15,14,10)(33,35)(34,40)(36,37)(38,46)(39,47)(41,42)(43,45)(44,48);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,16,10)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29)(33,36,39,46)(34,41,44,45)(35,37,47,38)(40,42,48,43);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,10,16,15)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)(33,37,39,38)(34,42,44,43)(35,36,47,46)(40,41,48,45)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 81075);