#96326

order	512
nilpotency class	3
isoclinism	isoclinic class of #7694
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

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	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=g,e^2=g,f^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,c^b=cg,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, 3522);