#8818123

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

4083 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)(19,21)(20,25)(22,23)(24,27)(26,29)(28,32)(30,31);(2,6)(4,13)(7,9)(11,16)(18,23)(21,24)(25,31)(29,32)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(2,6)(3,10)(8,12)(11,16)(17,19,22,27)(18,21,23,24)(20,28,30,26)(25,32,31,29)(33,34,37,40)(35,41,43,38)(36,39,45,44)(42,47,48,46);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,19)(18,24)(20,26)(21,23)(22,27)(25,32)(28,30)(29,31)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,22,30)(18,25,23,31)(19,26,27,28)(21,29,24,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21,22,24)(18,19,23,27)(20,29,30,32)(25,26,31,28)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)

Polycyclic presentation:

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

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, 8725319);