#174568

order	512
nilpotency class	3
isoclinism	isoclinic class of #174477
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

658 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgh,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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 81764);