#8720718

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

4007 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627914);