#8843850

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	#56386	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4123 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=h,d^2=gi,e^2=g,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, 8751046);