#6481894

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470409
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
prime radical	#29748	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1708 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)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18)(19,29)(20,34)(21,28)(22,27)(23,26)(24,32)(25,36)(30,42)(31,41)(33,40)(35,39)(37,44)(38,43)(45,46)(47,48);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,19,21,32,23,31,22,33)(18,24,27,29,26,40,28,41)(20,35,36,47,37,30,38,46)(25,42,34,48,43,39,44,45);(3,9)(6,13)(10,11)(14,15)(17,20)(18,25)(19,30)(21,36)(22,38)(23,37)(24,39)(26,43)(27,34)(28,44)(29,45)(31,35)(32,46)(33,47)(40,42)(41,48);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(19,31)(24,40)(29,41)(30,35)(32,33)(39,42)(45,48)(46,47);(2,8)(6,13)(7,16)(14,15)(18,26)(24,40)(25,43)(27,28)(29,41)(34,44)(39,42)(45,48)

Polycyclic presentation:

2^9|b^2=f,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,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, 6389090);