#6349845

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349830
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

2719 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)(3,11,13,15)(4,7,5,9)(8,14,16,12)(17,18,23,27)(19,28,32,41)(20,25,34,40)(21,26,22,36)(24,30,38,31)(29,42,44,48)(33,39,35,46)(37,43,47,45);(2,7)(4,5)(8,15)(9,10)(11,16)(12,14)(17,19,23,32)(18,24,27,38)(20,29,34,44)(21,31,22,30)(25,37,40,47)(26,41,36,28)(33,45,35,43)(39,48,46,42);(1,3)(2,8)(4,12)(5,14)(6,13)(7,15)(9,11)(10,16)(17,20)(18,25)(19,29)(21,33)(22,35)(23,34)(24,37)(26,39)(27,40)(28,42)(30,43)(31,45)(32,44)(36,46)(38,47)(41,48);(1,4,6,5)(2,9,10,7)(3,12,13,14)(8,11,16,15)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,33,34,35)(24,28,38,41)(25,39,40,46)(29,43,44,45)(37,42,47,48);(3,13)(8,16)(11,15)(12,14)(20,34)(25,40)(29,44)(33,35)(37,47)(39,46)(42,48)(43,45)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,d^b=dh,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, 6257041);