#121476

order	512
nilpotency class	3
isoclinism	isoclinic class of #10088
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

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	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

612 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:

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

Polycyclic presentation:

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

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, 28672);