#8814037

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	#50801	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3991 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:

(1,2,6,7)(3,5,10,8)(4,9,12,14)(11,13,16,15)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(33,34,39,36)(35,42,38,40)(37,41,45,44)(43,48,46,47);(2,7)(4,12)(5,8)(11,16)(17,18,22,19)(20,25,27,23)(21,26,29,24)(28,30,32,31)(37,45)(41,44)(43,46)(47,48);(2,7)(3,10)(4,12)(13,15)(18,19)(20,27)(24,26)(28,32)(33,35,39,38)(34,40,36,42)(37,43,45,46)(41,47,44,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,16)(13,14)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,36,39,34)(35,42,38,40)(37,44,45,41)(43,48,46,47);(1,4)(2,9)(3,11)(5,13)(6,12)(7,14)(8,15)(10,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,37,39,45)(34,41,36,44)(35,43,38,46)(40,47,42,48);(1,5,6,8)(2,3,7,10)(4,13,12,15)(9,11,14,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(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=i,c^2=ghi,d^2=gh,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, 8721233);