#8720839

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

4013 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)(17,18,23,29)(19,24,31,32)(20,26,30,25)(21,22,28,27)(34,38)(39,44)(40,45)(46,48);(1,2,5,6)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,19,23,31)(18,24,29,32)(20,22,30,27)(21,26,28,25)(35,41)(39,44)(42,47)(46,48);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,20,23,30)(18,25,29,26)(19,27,31,22)(21,24,28,32)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,20)(18,26)(19,27)(21,24)(22,31)(23,30)(25,29)(28,32)(33,35,37,41)(34,39,38,44)(36,42,43,47)(40,46,45,48);(1,3,5,9)(2,7,6,12)(4,10,11,15)(8,14,13,16)(17,21)(18,27)(19,26)(20,24)(22,29)(23,28)(25,31)(30,32)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,22,23,27)(18,28,29,21)(19,30,31,20)(24,25,32,26)(33,36,37,43)(34,40,38,45)(35,42,41,47)(39,46,44,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gi,d^2=hi,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, 8628035);