#121976

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#399
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2)(3,4)(5,6,8,11,9,7,12,10)(13,14)(15,24,27,40)(16,21)(17,22)(18,32)(19,23)(20,26,35,28)(25,36,42,41)(29,34,43,44)(30,37)(31,39)(33,38);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15,18,26,19,27,17,28)(14,20,32,24,23,35,22,40)(16,29,33,42,31,43,30,25)(21,36,38,34,39,41,37,44);(2,4)(6,10)(7,11)(8,12)(13,16,18,33,19,31,17,30)(14,21,32,38,23,39,22,37)(15,25,26,29,27,42,28,43)(20,34,24,41,35,44,40,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29172);