#6349291

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
prime radical	#60536	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2730 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)(4,9)(5,14)(6,8)(7,12)(11,15)(13,16)(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)(2,7,8,15)(4,12,5,11)(9,16,14,10)(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);(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,38)(26,41)(27,34)(28,42)(29,43)(31,45)(32,46)(33,47)(39,44)(40,48);(2,8)(7,15)(9,14)(10,16)(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)(2,9,8,14)(3,11,13,12)(7,10,15,16)(19,32)(24,39)(29,40)(30,46)(31,33)(38,44)(43,48)(45,47)

Polycyclic presentation:

2^9|b^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	#72306801, #72306805, #72306855
maximal normal subgroup of	#72306801, #72306805, #72306855

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256487);