#108463

order	512
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	C₄×C₂²⋊C₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

604 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 15659);