#121973

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,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, 29169);