#149296

order	512
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₂²×C₄	C₂²×C₈
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

478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,8)(4,6)(5,7)(9,10,15,22,16,21,38,51,14,20,36,50,13,19,35,49)(11,23,26,53,25,43,57,52,27,41,37,64,34,17,39,42)(12,28,33,59,32,46,62,69,31,18,44,47,30,48,61,71)(24,45,55,68,54,67,63,72,56,58,65,70,29,40,60,66);(1,3,5,4)(2,6,7,8)(9,11,16,25,14,27,13,34)(10,17,21,23,20,43,19,41)(12,29,32,24,31,54,30,56)(15,37,38,39,36,26,35,57)(18,45,48,67,28,58,46,40)(22,52,51,64,50,42,49,53)(33,63,62,65,44,60,61,55)(47,70,71,66,59,68,69,72);(1,4,5,3)(2,6,7,8)(9,12,16,32,14,31,13,30)(10,18,21,48,20,28,19,46)(11,24,25,54,27,56,34,29)(15,33,38,62,36,44,35,61)(17,40,23,45,43,67,41,58)(22,47,51,71,50,59,49,69)(26,55,57,63,37,65,39,60)(42,66,53,68,52,72,64,70)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=i,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=gi,f^c=fi

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, 56492);