#8864532

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

4046 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8771728);