#8726287

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

4119 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=h,d^2=gh,e^2=i,f^2=gi,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, 8633483);