#126896

order	512
nilpotency class	4
isoclinism	isoclinic class of #126723
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1736
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#7904
prime radical	C₂⁴	#104
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

466 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=g,e^2=h,f^2=gh,g^2=i,h^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fg,g^a=gi,c^b=cgh,e^b=ei,d^c=di,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214114, #72214259, #72214311, …
maximal normal subgroup of	#72214114, #72214259, #72214311, …
maximal quotient of	#72226839

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34092);