#9243647

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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9150843);