#105579

order	512
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

922 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12775);