#8317486

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

4005 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)(3,11)(4,14)(5,8)(6,12)(7,9)(10,15)(13,16)(33,34,39,38)(35,41,36,40)(37,42,46,45)(43,48,44,47);(1,3,7,5)(2,8,9,11)(4,12,15,13)(6,14,16,10)(17,18,22,23)(19,21,25,30)(20,24,29,31)(26,28,27,32)(33,35,39,36)(34,40,38,41)(37,43,46,44)(42,47,45,48);(2,9)(6,16)(8,11)(10,14)(18,23)(21,30)(24,31)(28,32)(33,35,39,36)(34,41,38,40)(37,43,46,44)(42,48,45,47);(1,4,7,15)(2,10,9,14)(3,13,5,12)(6,8,16,11)(17,19,22,25)(18,21,23,30)(20,27,29,26)(24,32,31,28)(33,36,39,35)(34,41,38,40)(37,43,46,44)(42,47,45,48);(1,5,7,3)(2,11,9,8)(4,13,15,12)(6,10,16,14)(17,20,22,29)(18,24,23,31)(19,26,25,27)(21,28,30,32)(33,37,39,46)(34,42,38,45)(35,43,36,44)(40,47,41,48);(1,6,7,16)(2,12,9,13)(3,14,5,10)(4,8,15,11)(17,21)(18,25)(19,23)(20,28)(22,30)(24,27)(26,31)(29,32)(33,38,39,34)(35,41,36,40)(37,45,46,42)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8224682);