#95027

order	512
nilpotency class	3
isoclinism	isoclinic class of #94978
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₄³	C₂³
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

516 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gi,d^2=gh,e^2=i,f^2=g,b^a=bd,d^a=dg,f^a=fgh,c^b=cgi,e^b=eh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2223);