#180303

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, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

670 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,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, 87499);