#8963667

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

4204 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8870863);