#8360919

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

4194 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,d^2=hi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8268115);