#496840

order	512
nilpotency class	4
isoclinism	isoclinic class of #483928
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	#3050	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

686 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)(3,7)(4,6)(5,8)(9,10,14,20,16,21,15,39)(11,17,25,43,23,41,26,60)(12,27,30,62,32,63,31,49)(13,33,36,50,38,67,37,19)(18,44,46,29,47,35,51,65)(22,52,55,71,53,72,28,61)(24,56,58,48,34,66,59,42)(40,64,45,54,69,57,70,68);(1,3)(2,5)(4,8)(6,7)(9,11,14,25,16,23,15,26)(10,17,20,43,21,41,39,60)(12,28,30,22,32,55,31,53)(13,34,36,59,38,24,37,58)(18,45,46,69,47,70,51,40)(19,48,33,66,50,42,67,56)(27,61,62,52,63,71,49,72)(29,57,35,68,65,64,44,54);(3,8)(5,7)(9,12,16,32)(10,18,21,47)(11,22,23,53)(13,35,38,44)(14,31,15,30)(17,40,41,69)(19,49,50,62)(20,51,39,46)(24,57,34,64)(25,28,26,55)(27,33,63,67)(29,37,65,36)(42,61,48,71)(43,70,60,45)(52,56,72,66)(54,59,68,58);(2,6)(5,7)(11,23)(17,41)(22,53)(24,34)(25,26)(28,55)(40,69)(42,48)(43,60)(45,70)(52,72)(54,68)(56,66)(57,64)(58,59)(61,71)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ceh,d^b=di,f^b=fi,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404036);