#8912662

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

4275 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8819858);