#121467

order	512
nilpotency class	3
isoclinism	isoclinic class of #10128
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

550 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28663);