#123954

order	512
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#386
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1634 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31150);