#123718

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

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,d^a=di,h^a=hi,c^b=cf,e^c=ei

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, 30914);