#8406750

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

4495 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8313946);