#9160790

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

4094 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)(3,7,11,13)(4,9,14,16)(5,12,15,10)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,31,30,32)(33,34,39,43)(35,44,42,38)(36,47,46,40)(37,41,45,48);(1,3)(2,7)(4,10)(5,9)(6,11)(8,13)(12,14)(15,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,31)(28,32)(29,30)(33,35,39,42)(34,38,43,44)(36,45,46,37)(40,48,47,41);(2,8)(3,11)(9,16)(10,12)(20,29)(24,28)(26,30)(31,32)(33,36,39,46)(34,40,43,47)(35,45,42,37)(38,48,44,41);(1,3)(2,7)(4,10)(5,9)(6,11)(8,13)(12,14)(15,16)(17,20,22,29)(18,24,21,28)(19,26,27,30)(23,31,25,32)(33,36)(34,40)(35,45)(37,42)(38,48)(39,46)(41,44)(43,47);(1,4,6,14)(2,9,8,16)(3,12,11,10)(5,7,15,13)(17,19)(18,25)(20,30)(21,23)(22,27)(24,31)(26,29)(28,32)(33,37,39,45)(34,41,43,48)(35,46,42,36)(38,47,44,40);(1,5,6,15)(2,10,8,12)(3,9,11,16)(4,13,14,7)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,32,30,31)(33,38)(34,42)(35,43)(36,48)(37,40)(39,44)(41,46)(45,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=hi,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=dgi,e^d=egi,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, 9067986);