#168086

order	512
nilpotency class	3
isoclinism	isoclinic class of #167761
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

696 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 75282);