#6351847

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351817
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	#62822	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2795 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)(3,11)(4,9)(5,8)(6,10)(7,13)(12,16)(14,15)(17,18,23,27)(19,28,32,42)(20,25,33,39)(21,26,22,35)(24,30,38,31)(29,44,45,43)(34,40,36,41)(37,47,48,46);(1,3)(2,7)(4,12)(5,13)(6,14)(8,15)(9,11)(10,16)(17,19)(18,24)(20,29)(21,31)(22,30)(23,32)(25,37)(26,42)(27,38)(28,35)(33,45)(34,47)(36,46)(39,48)(40,44)(41,43);(1,3)(2,7)(4,13)(5,12)(6,14)(8,11)(9,15)(10,16)(17,20)(18,25)(19,29)(21,34)(22,36)(23,33)(24,37)(26,40)(27,39)(28,43)(30,46)(31,47)(32,45)(35,41)(38,48)(42,44);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(20,33)(25,39)(29,45)(34,36)(37,48)(40,41)(43,44)(46,47);(1,4,6,5)(2,9,10,8)(3,13,14,12)(7,15,16,11)(17,21,23,22)(18,26,27,35)(19,30,32,31)(20,34,33,36)(24,28,38,42)(25,40,39,41)(29,46,45,47)(37,43,48,44)

Polycyclic presentation:

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

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, 6259043);