#8256586

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3980 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:

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

Polycyclic presentation:

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

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, 8163782);