#8873771

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

4122 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=hi,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, 8780967);