#123723

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	#380
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,h^a=hi,c^b=cf,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, 30919);