#8429524

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	#39256	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4107 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8336720);