#168998

order	512
nilpotency class	3
isoclinism	isoclinic class of #168909
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2679	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

859 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,d^2=h,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, 76194);