#146873

order	512
nilpotency class	4
isoclinism	isoclinic class of #11521
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂²×C₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂²	#1799
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

676 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

(5,6)(7,18,20,30)(8,21,23,12)(9,14)(10,26,28,15)(11,17)(13,22)(16,25)(19,33)(24,31,36,29)(27,34)(32,35);(1,2,3,4)(5,7,9,8,11,20,25,23)(6,12,16,30,17,21,14,18)(10,19,24,22,28,35,36,27)(13,31,33,26,34,29,32,15);(5,8,9,20,11,23,25,7)(6,13,14,32,17,34,16,33)(10,27,24,19,28,22,36,35)(12,29,18,26,21,31,30,15)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eh,g^a=gi,h^a=hi,c^b=ci,f^b=fi,e^c=ei,f^c=fhi,f^e=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54069);