#146546

order	512
nilpotency class	4
isoclinism	isoclinic class of #126723
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1736
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10730
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

943 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53742);