#120753

order	512
nilpotency class	3
isoclinism	isoclinic class of #10260
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#388
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

466 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,c^b=cf,d^b=dh,f^b=fi,g^b=gh,f^c=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27949);