#146846

order	512
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1746
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

470 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,g^a=gi,c^b=ch,d^b=di,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, 54042);