#179058

order	512
nilpotency class	3
isoclinism	isoclinic class of #178589
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

645 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)(11,15,18,24)(12,20,22,16)(14,21,23,19)(26,31)(28,29)(32,39)(33,37)(34,35)(38,40);(9,11)(10,14)(12,21)(13,18)(15,20)(16,24)(17,23)(19,22)(25,26,30,33)(27,32,36,40)(28,31,29,37)(34,39,35,38);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,19)(13,18)(14,20)(16,23)(17,24)(21,22)(25,27,30,36)(26,32,33,40)(28,35,29,34)(31,38,37,39);(1,3,5,4)(2,6,7,8)(11,18)(14,23)(15,24)(19,21)(25,28,30,29)(26,31,33,37)(27,34,36,35)(32,39,40,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=hi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eg,f^a=fi,d^b=dgh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 86254);