#146929

order	512
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1808
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(5,6,10,16,12,19,32,18)(7,20,23,49,25,50,57,37)(8,26,29,43,31,41,62,61)(9,17,11,15,33,46,34,47)(13,24,36,58,39,59,38,22)(14,40,42,27,45,30,44,28)(21,51,54,35,56,63,68,65)(48,66,67,52,60,55,64,53);(1,2,3,4)(5,7,11,24,10,23,33,58,12,25,34,59,32,57,9,22)(6,13,17,37,18,38,47,50,19,39,46,49,16,36,15,20)(8,21,30,55,29,54,28,53,31,56,40,66,62,68,27,52)(14,35,43,64,44,51,26,60,45,65,61,67,42,63,41,48);(5,8,9,27,32,62,34,40,12,31,33,28,10,29,11,30)(6,14,15,41,16,42,46,61,19,45,47,26,18,44,17,43)(7,21,22,52,57,68,59,66,25,56,58,53,23,54,24,55)(13,35,20,48,36,63,49,67,39,65,50,60,38,51,37,64)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=hi,e^2=hi,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,h^a=hi,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, 54125);