#8438526

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

4604 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,e^2=ghi,f^2=g,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, 8345722);