#180517

order	512
nilpotency class	3
isoclinism	isoclinic class of #180005
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	#2011	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

739 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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=hi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,d^b=dgi,e^b=eg,f^b=fh,d^c=dg,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, 87713);