#6349699

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349684
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	#62833	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2747 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)(19,23)(20,30)(21,33)(22,25)(24,32)(26,36)(27,43)(28,41)(29,38)(31,42)(34,39)(35,40)(37,45)(44,47)(46,48);(2,7)(4,5)(8,14)(9,10)(12,13)(15,16)(17,19,22,29)(18,23,25,38)(20,27,31,44)(21,34,36,28)(24,37,40,48)(26,41,33,39)(30,47,42,43)(32,46,35,45);(1,3)(2,8)(4,12)(5,13)(6,11)(7,14)(9,16)(10,15)(17,20)(18,24)(19,27)(21,35)(22,31)(23,37)(25,40)(26,42)(28,46)(29,44)(30,33)(32,36)(34,45)(38,48)(39,47)(41,43);(2,9)(3,11)(7,10)(12,13)(18,25)(23,38)(24,40)(26,33)(30,42)(37,48)(39,41)(43,47);(1,4,6,5)(2,7,9,10)(3,12,11,13)(8,14,16,15)

Polycyclic presentation:

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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256895);