#176513

order	512
nilpotency class	3
isoclinism	isoclinic class of #176333
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

646 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)(3,7)(6,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24)(25,26,31,27)(28,36,38,40)(29,33,30,35)(32,39,34,37);(1,2)(3,8)(4,6)(5,7)(9,11,13,19)(10,14,16,22)(12,18,20,24)(15,21,17,23)(25,27,31,26)(28,34,38,32)(29,35,30,33)(36,39,40,37);(10,15)(12,20)(14,21)(16,17)(18,24)(22,23)(25,28,31,38)(26,32,27,34)(29,37,30,39)(33,40,35,36);(2,6)(5,8)(10,16)(14,22)(15,17)(21,23)(25,29,31,30)(26,33,27,35)(28,37,38,39)(32,40,34,36)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,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=dg,e^b=eg,f^b=fh,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, 83709);