#123990

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#389
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1586 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)(9,10,13,19,15,20,14,18)(11,21,23,16,25,32,24,33)(12,26,29,17,30,35,28,36)(22,38,39,37,40,31,27,34);(2,4)(5,6)(7,8)(9,11,14,24,15,25,13,23)(10,16,19,32,20,33,18,21)(12,27,29,22,30,39,28,40)(17,34,26,31,36,37,35,38);(6,8)(9,12,13,29,15,30,14,28)(10,17,18,26,20,36,19,35)(11,22,23,39,25,40,24,27)(16,31,21,37,33,38,32,34)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^c=di,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, 31186);