#121305

order	512
nilpotency class	3
isoclinism	isoclinic class of #10637
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#397
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1443
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

389 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=i,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fi,g^b=ghi,e^c=ehi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28501);