#8197317

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	#51170	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=gh,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104513);