#8821841

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

3916 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=gh,e^2=ghi,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, 8729037);