#278514

order	512
nilpotency class	3
isoclinism	isoclinic class of #278491
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×AGL(1,ℤ/8)	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=eh,d^2=ehi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,g^b=gi,d^c=dh,f^c=fi,g^c=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185710);