#434691

order	512
nilpotency class	3
isoclinism	isoclinic class of #419852
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1034 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=ef,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^c=hi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341887);