#8416897

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

4230 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=i,e^2=gi,f^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, 8324093);