#8463260

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

4404 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=h,e^2=hi,f^2=i,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, 8370456);