#6477402

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470028
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	C₂²×C₄	C₂⁵
prime radical	#62814	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2950 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

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, 6384598);