#106914

order	512
nilpotency class	3
isoclinism	isoclinic class of #9159
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

913 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

(2,5,6,7)(9,10,13,18,15,12,14,17)(11,19,21,16,23,24,22,20);(1,2)(3,7)(4,6)(5,8)(9,11,14,22,15,23,13,21)(10,16,18,24,12,20,17,19);(9,12)(10,15)(11,20)(13,18)(14,17)(16,23)(19,22)(21,24)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=di,f^b=fi,e^c=eh,f^c=fh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14110);