#470583

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1078 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,d^c=de,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377779);