#173369

order	512
nilpotency class	3
isoclinism	isoclinic class of #172777
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	#2679	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

847 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)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,21,24)(12,16,13,22)(15,19,23,20)(25,26,30,34)(27,35,36,39)(28,33,29,32)(31,38,40,37);(2,5)(3,8)(6,7)(9,11)(10,15)(12,20)(13,19)(14,21)(16,24)(17,23)(18,22)(25,27)(26,31)(28,38)(29,37)(30,36)(32,39)(33,35)(34,40);(9,10)(11,15)(12,16)(13,22)(14,17)(18,19)(20,24)(21,23)(26,32)(28,29)(31,39)(33,34)(35,40)(37,38);(2,6)(5,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,18,23,24)(27,36)(31,40)(35,39)(37,38)

Polycyclic presentation:

2^9|a^2=h,d^2=hi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fi,d^b=dh,e^b=eg,f^b=fh,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, 80565);