#105105

order	512
nilpotency class	3
isoclinism	isoclinic class of #105073
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
commutator, Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

421 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=d,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,c^b=cf,d^b=dg,e^b=ei,f^b=fi,e^c=egh,f^c=fgi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12301);