#6349825

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349761
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	#62759	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2801 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257021);