#126967

order	512
nilpotency class	3
isoclinism	isoclinic class of #8101
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#95
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

472 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)(9,10,13,19,15,20,27,36,14,18,29,37,26,35,12,17)(11,16,23,31,24,33,25,34,22,32,28,38,39,40,21,30);(2,5)(3,7)(9,11)(10,16,18,32)(12,25)(13,28)(14,22)(15,24)(17,34,36,30)(19,38,37,31)(20,33,35,40)(21,27)(23,29)(26,39);(9,12,26,29,14,27,15,13)(10,17,35,37,18,36,20,19)(11,21,39,28,22,25,24,23)(16,30,40,38,32,34,33,31)

Polycyclic presentation:

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

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, 34163);