#180156

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	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

706 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)(9,10,13,17)(11,14,18,22)(12,15,20,16)(19,21,24,23)(26,31)(27,34)(28,29)(33,36)(35,37)(38,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,21)(16,23)(17,22)(20,24)(25,26,30,33)(27,32,35,39)(28,36,29,31)(34,40,37,38);(3,4)(6,7)(10,15)(12,20)(14,21)(16,17)(19,24)(22,23)(25,27)(26,32)(28,34)(29,37)(30,35)(31,38)(33,39)(36,40);(1,3,5,4)(2,7,8,6)(11,18)(14,22)(19,24)(21,23)(27,35)(32,39)(34,37)(38,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,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, 87352);