#146979

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	#1955
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

544 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)(5,6,10,17,12,19,33,47,13,20,11,18,9,16,32,46)(7,21,25,50,23,38,57,48,24,44,34,61,30,14,36,39)(8,15,28,42,27,41,60,68,29,43,35,49,31,45,52,62)(22,51,55,67,26,37,58,64,54,40,56,66,53,63,59,65);(2,4)(5,7,11,25,12,23,32,57,13,24,10,34,9,30,33,36)(6,14,18,39,19,21,46,50,20,38,17,48,16,44,47,61)(8,26,28,59,27,54,60,55,29,53,35,58,31,22,52,56)(15,40,42,67,41,63,68,64,43,51,49,66,45,37,62,65);(5,8)(6,15)(7,22,24,54)(9,31)(10,28)(11,35)(12,27)(13,29)(14,37,38,63)(16,45)(17,42)(18,49)(19,41)(20,43)(21,40,44,51)(23,26,30,53)(25,55,34,56)(32,52)(33,60)(36,59,57,58)(39,64,48,65)(46,62)(47,68)(50,66,61,67)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,d^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=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, 54175);