#106745

order	512
nilpotency class	3
isoclinism	isoclinic class of #106738
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1760 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13941);