#8745818

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

4026 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

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, 8653014);