#496064

order	512
nilpotency class	4
isoclinism	isoclinic class of #483962
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

543 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=e,d^2=gi,e^2=h,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,d^b=dhi,g^b=gi,d^c=dh,g^c=gi,e^d=ei,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403260);