#8734965

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

4655 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=h,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8642161);