#8197343

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

4593 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)(3,7)(4,6)(5,8)(10,15)(11,12)(18,22)(19,20)(25,26)(27,37)(28,34)(29,39)(30,35)(31,33)(32,36)(38,40);(1,2,4,6)(3,5,8,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,27,32,31)(26,33,36,37)(28,38,29,35)(30,39,40,34);(1,2)(3,5)(4,6)(7,8)(9,11,14,12)(10,16,15,17)(13,20,21,19)(18,24,22,23)(25,28,32,29)(26,34,36,39)(27,38,31,35)(30,37,40,33);(1,3,4,8)(2,5,6,7)(9,12,14,11)(10,17,15,16)(13,20,21,19)(18,24,22,23)(25,29,32,28)(26,34,36,39)(27,38,31,35)(30,33,40,37);(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,30,32,40)(26,35,36,38)(27,39,31,34)(28,37,29,33);(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,31,32,27)(26,33,36,37)(28,38,29,35)(30,34,40,39)

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=g,e^2=ghi,f^2=ghi,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, 8104539);