#6351818

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

Polycyclic presentation:

2^9|a^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, 6259014);