#8898919

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

4186 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)(3,10)(4,5)(6,7)(8,13)(9,15)(11,12)(14,16)(17,18)(19,21)(20,25)(22,31)(23,24)(26,30)(27,28)(29,32)(33,34,39,38)(35,41,36,40)(37,42,46,45)(43,48,44,47);(2,7)(4,9)(8,10)(11,14)(17,19,23,22)(18,21,24,31)(20,26,29,28)(25,30,32,27)(33,35,39,36)(34,40,38,41)(37,43,46,44)(42,47,45,48);(2,7)(5,15)(8,10)(12,16)(18,24)(19,22)(25,32)(26,28)(33,35,39,36)(34,41,38,40)(37,43,46,44)(42,48,45,47);(1,3,6,13)(2,8,7,10)(4,14,9,11)(5,12,15,16)(17,20)(18,25)(19,26)(21,30)(22,28)(23,29)(24,32)(27,31)(33,36,39,35)(34,41,38,40)(37,43,46,44)(42,47,45,48);(1,4,6,9)(2,5,7,15)(3,11,13,14)(8,16,10,12)(17,21)(18,19)(20,27)(22,24)(23,31)(25,28)(26,32)(29,30)(33,37,39,46)(34,42,38,45)(35,43,36,44)(40,47,41,48);(1,5)(2,9)(3,12)(4,7)(6,15)(8,11)(10,14)(13,16)(17,22,23,19)(18,21,24,31)(20,28,29,26)(25,30,32,27)(33,38,39,34)(35,41,36,40)(37,45,46,42)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=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, 8806115);