#121457

order	512
nilpotency class	3
isoclinism	isoclinic class of #10120
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

522 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)(5,6,11,14)(7,15,18,16)(8,9,13,10)(12,19,20,17)(21,22)(23,27)(24,26)(25,32)(28,31)(29,35)(30,34)(33,36);(5,7,9,17,11,18,10,19)(6,12,8,15,14,20,13,16)(22,26)(23,29,30,33)(27,35,34,36)(28,32);(5,8,10,6,11,13,9,14)(7,16,17,12,18,15,19,20)(21,23)(22,27)(24,30)(25,33)(26,34)(28,35)(29,31)(32,36)

Polycyclic presentation:

2^9|a^2=g,b^2=di,c^2=e,d^2=h,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,e^b=eh,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, 28653);