#10143476

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

4241 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=i,d^2=ghi,e^2=h,f^2=gh,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, 10050672);