#123409

order	512
nilpotency class	4
isoclinism	isoclinic class of #123403
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₁₆

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1712
commutator, Frattini, cosocle	C₈²	C₂³
prime radical	#1712	C₂²
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

229 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(2,9)(3,14)(4,5)(6,7)(10,15)(11,13)(12,16)(17,18)(19,27)(20,32)(21,24)(22,35)(23,29)(25,28)(26,33)(30,39)(31,46)(34,47)(36,45)(37,48)(38,43)(40,42)(41,44);(1,2,10,16,6,12,4,9,8,14,5,11,7,13,15,3)(17,19,28,42,22,31,34,45,21,30,43,48,33,44,20,29)(18,23,32,41,26,37,38,39,24,36,47,46,35,40,25,27);(1,3,15,13,7,11,5,14,8,9,4,12,6,16,10,2)(18,24)(23,36)(25,38)(26,35)(27,39)(32,47)(37,40)(41,46)

Polycyclic presentation:

2^9|b^2=dfi,c^2=egh,d^2=fh,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gi

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225191, #72225197, #72225205
maximal normal subgroup of	#72225191, #72225197, #72225205

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30605);