#8943945

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

4236 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=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, 8851141);