#127153

order	512
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁴	C₄×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

448 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,e^2=h,f^2=gh,h^2=i,b^a=bd,c^a=ci,d^a=dg,e^b=eg

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34349);