#8814027

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4118 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=hi,d^2=h,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,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, 8721223);