#8889446

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

4099 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)(3,9)(4,10)(5,13)(6,7)(8,14)(11,12)(15,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32)(33,34,39,38)(35,40,44,43)(36,45,47,41)(37,48,46,42);(2,7)(4,9)(5,14)(11,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32)(34,38)(40,43)(41,45)(42,48);(2,7)(3,10)(8,13)(11,16)(18,22)(23,25)(24,28)(30,31)(33,35,39,44)(34,40,38,43)(36,46,47,37)(41,48,45,42);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,36,39,47)(34,41,38,45)(35,37,44,46)(40,42,43,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,37)(34,42)(35,36)(38,48)(39,46)(40,41)(43,45)(44,47);(1,5,6,14)(2,8,7,13)(3,11,10,16)(4,12,9,15)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,38,39,34)(35,43,44,40)(36,45,47,41)(37,48,46,42)

Polycyclic presentation:

2^9|a^2=ghi,c^2=ghi,d^2=g,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8796642);