#8429394

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3959 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)(17,18,22,21)(19,23,25,27)(20,24,28,29)(26,30,31,32)(34,36)(39,41)(40,44)(46,47);(1,2,6,4)(3,10,9,7)(5,8,14,12)(11,16,13,15)(19,25)(23,27)(26,31)(30,32)(33,34,38,36)(35,39,43,41)(37,40,45,44)(42,46,48,47);(1,2,6,4)(3,7,9,10)(5,8,14,12)(11,15,13,16)(17,19,22,25)(18,23,21,27)(20,26,28,31)(24,30,29,32)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,3,6,9)(2,7,4,10)(5,13,14,11)(8,16,12,15)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,36,38,34)(35,41,43,39)(37,40,45,44)(42,46,48,47);(1,4,6,2)(3,10,9,7)(5,12,14,8)(11,16,13,15);(1,5,6,14)(2,8,4,12)(3,11,9,13)(7,15,10,16)(17,21,22,18)(19,27,25,23)(20,24,28,29)(26,30,31,32)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=g,e^2=h,f^2=gi,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, 8336590);