#172166

order	512
nilpotency class	3
isoclinism	isoclinic class of #172137
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

799 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,d^b=dgh,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, 79362);