#6498674

order	512
nilpotency class	3
isoclinism	isoclinic class of #6473153
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	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1683 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)(4,11)(5,7)(8,12)(9,13)(10,16)(14,15)(17,18,21,27,23,26,22,38)(19,28,31,40,30,41,37,24)(20,32,34,43,36,46,35,25)(29,39,45,44,33,42,47,48);(1,3)(2,6)(4,10)(5,9)(7,13)(8,14)(11,16)(12,15)(17,19)(18,24)(20,33)(21,37)(22,31)(23,30)(25,42)(26,40)(27,41)(28,38)(29,36)(32,44)(34,45)(35,47)(39,43)(46,48);(2,7)(3,9)(8,11)(10,15)(17,20,21,34,23,36,22,35)(18,25,38,46,26,43,27,32)(19,29,37,47,30,33,31,45)(24,39,28,44,40,42,41,48);(3,9)(6,13)(10,15)(14,16)(19,30)(24,40)(28,41)(29,33)(31,37)(39,42)(44,48)(45,47);(1,4,5,12)(2,8,7,11)(3,10,9,15)(6,14,13,16)(18,26)(24,40)(25,43)(27,38)(28,41)(32,46)(39,42)(44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6405870);