#8718965

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

4500 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8626161);