#168000

order	512
nilpotency class	3
isoclinism	isoclinic class of #167761
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

812 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=i,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, 75196);