#8665309

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

4600 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)(11,17)(12,21)(13,14)(15,20)(16,22)(18,23)(19,24)(26,31)(32,33)(34,37)(39,40);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,23)(19,22)(21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,3,5,6)(2,4,7,8)(9,12)(10,16)(11,18)(13,21)(14,22)(15,23)(17,24)(19,20)(25,27,30,28)(26,32,31,33)(29,35,38,36)(34,39,37,40);(1,2)(3,4)(5,7)(6,8)(9,12)(10,16)(11,19)(13,21)(14,22)(15,24)(17,23)(18,20)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,40)(36,39);(1,2)(3,4)(5,7)(6,8)(9,13,14,10)(11,17,20,15)(12,21,22,16)(18,24,19,23)(25,28,30,27)(26,33,31,32)(29,35,38,36)(34,39,37,40);(1,4)(2,6)(3,7)(5,8)(9,11)(10,17)(12,18)(13,15)(14,20)(16,24)(19,22)(21,23)(25,29,30,38)(26,34,31,37)(27,35,28,36)(32,39,33,40)

Polycyclic presentation:

2^9|c^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8572505);