#151125

order	512
nilpotency class	5
isoclinism	isoclinic class of #13697
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₁₆
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
socle, prime radical	C₂²	C₄×D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=e,d^2=fh,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=cgi,d^a=df,f^a=fh,h^a=hi,c^b=cg,d^b=df,f^b=fh,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58321);