#338948

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=ehi,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=deh,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 246144);