#6349293

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349215
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,10)(4,8)(5,14)(6,9)(7,11)(12,16)(13,15)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,33)(25,35)(30,44)(31,39)(32,40)(36,42)(37,41)(38,46)(43,47)(45,48);(1,3,6,13)(2,7,9,16)(4,11,5,12)(8,15,14,10)(17,19,23,32)(18,24,28,39)(20,30,37,46)(21,31,22,33)(25,38,42,44)(26,29,27,40)(34,48,41,43)(35,45,36,47);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,16,10)(17,20,23,37)(18,25,28,42)(19,30,32,46)(21,35,22,36)(24,38,39,44)(26,41,27,34)(29,43,40,48)(31,45,33,47);(2,9)(7,16)(8,14)(10,15)(17,21,23,22)(18,26,28,27)(19,31,32,33)(20,35,37,36)(24,29,39,40)(25,41,42,34)(30,45,46,47)(38,43,44,48);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,10,16,15)(19,32)(24,39)(29,40)(30,46)(31,33)(38,44)(43,48)(45,47)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306814, #72306819, #72306855
maximal normal subgroup of	#72306814, #72306819, #72306855

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256489);