#149108

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₈×D₄
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

462 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,e^2=i,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56304);