#6355015

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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1632 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)(3,7)(4,9)(5,14)(6,10)(8,11)(12,16)(13,15)(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)(2,7)(4,13)(5,11)(6,12)(8,14)(9,15)(10,16)(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);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,16)(10,15)(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);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,8,16,15)(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=hi,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, 6262211);