#96337

order	512
nilpotency class	3
isoclinism	isoclinic class of #7694
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

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

540 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=hi,d^2=g,e^2=g,f^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,c^b=cgh,d^b=dh

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, 3533);