#123678

order	512
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#392
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1490 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)(9,10,13,18,15,19,14,28)(11,20,24,16,23,31,22,30)(12,17,25,33,27,34,26,40)(21,36,39,32,38,35,37,29);(5,6)(7,8)(9,11,14,24,15,23,13,22)(10,16,18,20,19,30,28,31)(12,21,26,39,27,38,25,37)(17,32,33,36,34,29,40,35);(2,4)(6,8)(9,12,13,25,15,27,14,26)(10,17,18,33,19,34,28,40)(11,21,22,37,23,38,24,39)(16,29,20,35,30,32,31,36)

Polycyclic presentation:

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

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, 30874);