#108503

order	512
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#117
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1434
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

568 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,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, 15699);