#146831

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1832
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

454 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)(5,6,10,16,11,17,34,49,12,18,35,50,9,15,31,46)(7,19,23,51,21,40,55,47,24,38,32,63,22,13,36,39)(8,25,29,58,28,45,61,48,30,43,33,64,27,14,41,44)(20,52,53,66,26,59,60,68,54,37,57,65,56,42,62,67);(5,7)(6,13)(8,26,30,56)(9,22)(10,32)(11,21)(12,24)(14,42,45,59)(15,38)(16,47)(17,19)(18,40)(20,28,54,27)(23,35)(25,52,43,37)(29,62,33,60)(31,55)(34,36)(39,50)(41,57,61,53)(44,68,48,67)(46,51)(49,63)(58,65,64,66);(2,4)(5,8,9,27,12,30,11,28)(6,14,15,43,18,45,17,25)(7,20,21,26,24,54,22,56)(10,33,31,61,35,29,34,41)(13,37,19,42,40,52,38,59)(16,48,46,58,50,44,49,64)(23,57,55,62,32,53,36,60)(39,66,51,68,47,65,63,67)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,c^b=ch,d^b=di,f^b=fi,e^c=ei,f^c=fi,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, 54027);