#8889383

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

4095 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=g,e^2=i,f^2=gi,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, 8796579);