#147903

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

948 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,4)(5,6)(7,8)(9,10,15,22,16,20,39,23)(11,24,28,52,29,55,40,44)(12,30,34,51,35,18,41,50)(13,21,36,19,37,54,14,38)(17,27,45,26,43,62,46,63)(25,47,60,68,61,57,64,72)(31,42,59,70,65,69,67,71)(32,53,58,56,66,49,33,48);(2,4)(9,11,16,29)(10,17,20,43)(12,31,35,65)(13,27,37,62)(14,26,36,63)(15,40,39,28)(18,47,30,57)(19,52,38,44)(21,55,54,24)(22,46,23,45)(25,58,61,33)(32,64,66,60)(34,67,41,59)(42,53,69,49)(48,71,56,70)(50,72,51,68);(6,8)(9,12,16,35)(10,18,20,30)(11,25,29,61)(13,33,37,58)(14,32,36,66)(15,41,39,34)(17,42,43,69)(19,53,38,49)(21,56,54,48)(22,51,23,50)(24,57,55,47)(26,31,63,65)(27,59,62,67)(28,64,40,60)(44,72,52,68)(45,71,46,70)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,f^b=fh,g^b=gi,h^b=hi,d^c=dhi,f^c=fh,g^c=gi,h^c=hi

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, 55099);