#121818

order	512
nilpotency class	3
isoclinism	isoclinic class of #10645
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#397
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

406 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)(6,10)(7,14)(8,16)(11,19)(12,13)(15,20)(21,22)(23,30)(24,27)(25,29)(26,32)(28,34)(31,35)(33,36);(5,6,8,10,9,13,16,12)(7,15,18,20,14,19,17,11)(21,23,25,33)(22,26,29,35)(24,31,34,32)(27,30,28,36);(5,7,16,17,9,14,8,18)(6,11,12,19,13,20,10,15)(22,27)(23,31)(24,34)(26,35)(28,29)(32,33)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=deh,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=di,e^b=ei,f^b=fh,d^c=dh,e^c=eh,f^c=fh

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, 29014);