#9230884

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

4168 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=h,e^2=ghi,f^2=gh,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, 9138080);