#8806855

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

3910 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)(3,7)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(18,22)(19,27)(20,30)(26,32)(33,34,39,36)(35,42,38,40)(37,41,45,44)(43,48,46,47);(1,2)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18,23,22)(19,28,27,24)(20,25,30,31)(21,32,29,26)(37,45)(41,44)(43,46)(47,48);(2,6)(3,10)(8,12)(11,16)(17,19,23,27)(18,24,22,28)(20,29,30,21)(25,32,31,26)(33,35,39,38)(34,40,36,42)(37,43,45,46)(41,47,44,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20)(18,25)(19,29)(21,27)(22,31)(23,30)(24,32)(26,28)(33,36,39,34)(35,42,38,40)(37,44,45,41)(43,48,46,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,21,23,29)(18,26,22,32)(19,20,27,30)(24,25,28,31)(33,37,39,45)(34,41,36,44)(35,43,38,46)(40,47,42,48);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,22,23,18)(19,28,27,24)(20,31,30,25)(21,32,29,26)(33,38,39,35)(34,42,36,40)(37,46,45,43)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,d^2=hi,e^2=ghi,f^2=h,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, 8714051);