#123682

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	#399
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)(11,21,24,34)(12,25,27,35)(13,18)(14,19)(15,20)(16,23,31,28)(17,26,33,29)(22,30)(32,39)(36,37)(38,40);(5,6)(7,8)(9,11)(10,16,20,31)(12,22)(13,23)(14,28)(15,24)(17,32,33,30)(18,21,19,34)(25,40,35,36)(26,37)(27,39)(29,38);(2,4)(6,8)(9,12,13,26,15,27,14,29)(10,17,18,25,20,33,19,35)(11,22,23,37,24,39,28,38)(16,30,21,36,31,32,34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=i,b^a=bd,c^a=ce,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^b=gi,e^c=ei,g^c=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, 30878);