#8184868

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

4618 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)(3,4)(5,6)(7,8)(9,10,16,15)(11,13,17,12)(14,22,23,18)(19,20,24,21)(25,26)(27,32)(28,34)(29,33)(30,35)(31,36)(37,38)(39,40);(2,6)(3,7)(9,11)(10,13)(12,15)(14,19)(16,17)(18,21)(20,22)(23,24)(25,27,31,29)(26,32,36,33)(28,39,35,37)(30,38,34,40);(1,3,5,7)(2,4,6,8)(9,12,16,13)(10,17,15,11)(14,20,23,21)(18,24,22,19)(25,27,31,29)(26,33,36,32)(28,37,35,39)(30,38,34,40);(1,4)(2,7)(3,6)(5,8)(9,13,16,12)(10,17,15,11)(14,21,23,20)(18,24,22,19)(25,28,31,35)(26,34,36,30)(27,37,29,39)(32,38,33,40);(9,14,16,23)(10,18,15,22)(11,19,17,24)(12,20,13,21)(25,29,31,27)(26,32,36,33)(28,39,35,37)(30,40,34,38);(9,15,16,10)(11,12,17,13)(14,22,23,18)(19,20,24,21)(25,30,31,34)(26,35,36,28)(27,38,29,40)(32,37,33,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8092064);