#8445186

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

4590 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8352382);