#8406700

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

4335 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:

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

Polycyclic presentation:

2^9|a^2=g,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, 8313896);