#149183

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

550 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:

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

Polycyclic presentation:

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

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, 56379);