#8864543

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

4005 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=gi,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, 8771739);