#6349417

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349370
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	#62830	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2770 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:

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

Polycyclic presentation:

2^9|a^2=h,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,d^c=dh,f^c=fi,g^c=gi

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, 6256613);