#8096531

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

4581 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)(11,18,14,16)(12,21,23,13)(19,20,22,24)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(1,2)(3,6)(4,7)(5,8)(9,11,15,14)(10,16,17,18)(12,19,23,22)(13,20,21,24)(25,27,32,30)(26,33,29,28)(31,35,39,38)(34,40,37,36);(1,3)(2,5)(4,8)(6,7)(25,28)(26,30)(27,29)(31,36)(32,33)(34,38)(35,37)(39,40);(9,12)(10,13)(11,19)(14,22)(15,23)(16,20)(17,21)(18,24)(25,29,32,26)(27,33,30,28)(31,37,39,34)(35,40,38,36);(1,3)(2,6)(4,8)(5,7)(9,13)(10,12)(11,20)(14,24)(15,21)(16,19)(17,23)(18,22)(25,30,32,27)(26,33,29,28)(31,35,39,38)(34,36,37,40);(1,2)(3,5)(4,7)(6,8)(9,14,15,11)(10,16,17,18)(12,22,23,19)(13,20,21,24)(25,31,32,39)(26,34,29,37)(27,35,30,38)(28,36,33,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=h,e^2=h,f^2=g,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, 8003727);