#180293

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

671 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)(10,14)(12,19)(15,21)(16,20)(18,23)(22,24)(25,26,31,28)(27,35,39,40)(29,34,30,33)(32,37,36,38);(3,4)(6,8)(9,10)(11,15)(12,20)(13,16)(14,19)(17,22)(18,24)(21,23)(25,27)(26,32)(28,36)(29,38)(30,37)(31,39)(33,35)(34,40);(1,2,5,7)(3,6,4,8)(9,11,13,17)(10,15,16,22)(12,18,19,23)(14,21,20,24)(25,28,31,26)(27,36,39,32)(29,33,30,34)(35,37,40,38);(1,3,5,4)(2,6,7,8)(10,16)(11,17)(14,20)(18,23)(25,29,31,30)(26,33,28,34)(27,37,39,38)(32,40,36,35)

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,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, 87489);