#121288

order	512
nilpotency class	3
isoclinism	isoclinic class of #121286
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1443
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

388 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=i,e^2=i,f^2=i,g^2=h,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=fh,g^b=ghi,e^c=ehi,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28484);