#6354986

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354981
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60940	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1700 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)(3,6)(4,7)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,31)(29,43)(30,39)(32,38)(33,40)(34,41)(35,42)(37,45)(44,47)(46,48);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,19,22,31,23,32,21,30)(18,24,26,28,27,39,36,38)(20,29,34,45,35,46,33,44)(25,37,40,43,42,47,41,48);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20)(18,25)(19,29)(21,33)(22,34)(23,35)(24,37)(26,40)(27,42)(28,43)(30,44)(31,45)(32,46)(36,41)(38,48)(39,47);(3,10)(6,14)(11,16)(13,15)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,33,35,34)(24,28,39,38)(25,40,42,41)(29,44,46,45)(37,43,47,48);(17,21,23,22)(18,26,27,36)(19,31,32,30)(20,34,35,33)(24,38,39,28)(25,41,42,40)(29,44,46,45)(37,43,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262182);