#8968122

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

4418 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=hi,e^2=h,f^2=hi,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, 8875318);