#8534551

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

3950 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ghi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8441747);