#94860

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₄³	C₂³
commutator, socle, prime radical	C₂³	C₄³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

458 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,f^2=i,b^a=bgi,c^a=cgh,c^b=cg

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2056);