#8118586

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3926 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8025782);