#9214726

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

4162 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,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, 9121922);