#123654

order	512
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#117
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#382
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1474 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)(11,19,22,32)(12,23,25,16)(13,17)(14,18)(15,21,28,36)(20,33)(24,38,39,29)(26,30)(27,34)(31,35)(37,40);(5,6)(7,8)(9,11)(10,15,18,28)(12,20)(13,21)(14,22)(16,27,23,37)(17,19,30,32)(24,34)(25,35)(26,36)(29,31,38,33)(39,40);(2,4)(6,8)(9,12,13,24,14,25,26,39)(10,16,17,29,18,23,30,38)(11,20,21,34,22,35,36,40)(15,27,19,31,28,37,32,33)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,b^a=bd,c^a=ce,e^a=ei,h^a=hi,c^b=cf,g^b=gi,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, 30850);