#8096254

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

4439 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003450);