#8873773

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	#46333	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4017 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)(3,4)(5,13)(6,7)(8,9)(10,14)(11,15)(12,16)(17,18,22,19)(20,25,28,23)(21,26,29,24)(27,30,32,31)(33,34)(35,37)(36,44)(38,39)(40,43)(41,47)(42,48)(45,46);(2,7)(4,8)(5,14)(11,16)(17,18,22,19)(20,25,28,23)(21,26,29,24)(27,30,32,31)(34,39)(37,40)(41,44)(46,48);(2,7)(3,9)(10,13)(11,16)(18,19)(23,25)(24,26)(30,31)(33,35,38,43)(34,37,39,40)(36,42,47,45)(41,46,44,48);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,13)(14,16)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31)(33,35)(34,40)(36,45)(37,39)(38,43)(41,46)(42,47)(44,48);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,36,38,47)(34,41,39,44)(35,42,43,45)(37,46,40,48);(1,5,6,14)(2,10,7,13)(3,11,9,16)(4,12,8,15)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,40)(34,35,39,43)(36,46,47,48)(41,42,44,45)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=i,e^2=gi,f^2=g,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, 8780969);