#99430

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#385
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

548 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=gh,d^2=g,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg,e^b=eh,d^c=dh,e^c=eg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6626);