#10159216

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

4050 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18)(19,21)(20,24)(22,26)(23,30)(25,31)(27,32)(28,29)(33,34,40,44)(35,43,39,38)(36,45,47,41)(37,46,48,42);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,22,25)(18,21,26,31)(20,28,30,27)(23,32,24,29)(33,35,40,39)(34,38,44,43)(36,42,47,46)(37,45,48,41);(2,3)(7,9)(8,10)(14,15)(17,20,22,30)(18,23,26,24)(19,27,25,28)(21,29,31,32)(33,36,40,47)(34,41,44,45)(35,42,39,46)(37,43,48,38);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,24)(19,28)(21,29)(22,30)(23,26)(25,27)(31,32)(33,37,40,48)(34,42,44,46)(35,45,39,41)(36,38,47,43);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,19)(18,21)(20,27)(22,25)(23,29)(24,32)(26,31)(28,30)(33,38)(34,35)(36,37)(39,44)(40,43)(41,42)(45,46)(47,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21)(18,25)(19,26)(20,29)(22,31)(23,28)(24,27)(30,32)(33,39,40,35)(34,43,44,38)(36,46,47,42)(37,41,48,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10066412);