#118937

order	512
nilpotency class	3
isoclinism	isoclinic class of #10645
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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

514 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 52 points:

(5,6)(7,15)(8,12)(9,19)(10,14)(11,16)(13,18)(17,20)(21,22,26,31,27,30,25,41)(23,32,36,47,34,28,35,43)(24,29,40,46,38,45,39,48)(33,42,50,44,51,49,37,52);(1,2)(3,4)(5,7,10,17)(6,11,14,12)(8,18,16,9)(13,20,19,15)(21,23)(22,28)(24,37)(25,36)(26,35)(27,34)(29,44)(30,32)(31,47)(33,39)(38,50)(40,51)(41,43)(42,46)(45,52)(48,49);(2,4)(5,8,9,7,10,16,18,17)(6,12,13,20,14,11,19,15)(21,24)(22,29,30,45)(23,33)(25,40)(26,39)(27,38)(28,42,32,49)(31,48,41,46)(34,51)(35,50)(36,37)(43,52,47,44)

Polycyclic presentation:

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

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, 26133);