#178716

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

714 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)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24)(25,26,30,32)(27,35,36,40)(28,34,29,33)(31,38,39,37);(1,2,5,7)(3,6,4,8)(9,11,13,19)(10,14,16,22)(12,18,20,24)(15,21,17,23)(25,27)(26,31)(28,38)(29,37)(30,36)(32,39)(33,40)(34,35);(3,4)(6,8)(10,15)(12,20)(14,21)(16,17)(18,24)(22,23)(25,27)(26,31)(28,37)(29,38)(30,36)(32,39)(33,35)(34,40);(1,3,5,4)(2,6,7,8)(10,16)(14,22)(15,17)(21,23)(26,32)(27,36)(33,34)(37,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=h,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, 85912);