#106612

order	512
nilpotency class	3
isoclinism	isoclinic class of #8346
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1050 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=di,e^b=eh,f^b=fhi,d^c=dh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287524, #72318965, #72343021, …
maximal normal subgroup of	#72287524, #72318965, #72343021, …
maximal quotient of	#72287583

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13808);