#167848

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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

760 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,b^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, 75044);