#180134

order	512
nilpotency class	3
isoclinism	isoclinic class of #180005
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

665 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)(10,15)(12,13)(16,23)(17,18)(19,20)(22,24)(25,26,29,33)(27,30,34,38)(28,31,36,32)(35,37,40,39);(3,4)(6,8)(9,10)(11,16)(12,15)(13,17)(14,18)(19,23)(20,22)(21,24)(25,27)(26,30)(28,35)(29,34)(31,37)(32,39)(33,38)(36,40);(1,2)(3,6)(4,8)(5,7)(9,11,14,21)(10,16,18,24)(12,20,13,19)(15,22,17,23)(26,31)(28,36)(30,37)(32,33)(35,40)(38,39);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,17,18,15)(11,19,21,20)(16,22,24,23)(27,34)(30,38)(35,40)(37,39)

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,d^2=h,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, 87330);