#107336

order	512
nilpotency class	3
isoclinism	isoclinic class of #107334
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

898 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14532);