#96385

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	C₄³	C₂³
socle	C₂³	C₂×#95
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gi,d^2=g,e^2=h,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, 3581);