#8188492

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

4410 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)(4,5)(6,8)(10,15)(11,12)(18,22)(19,20)(26,31)(32,33)(34,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,26,30,31)(27,32,28,33)(29,34,37,38)(35,39,36,40);(2,5)(6,7)(9,11,14,12)(10,16,15,17)(13,20,21,19)(18,24,22,23)(25,27,30,28)(26,32,31,33)(29,36,37,35)(34,40,38,39);(9,12,14,11)(10,17,15,16)(13,20,21,19)(18,24,22,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,3,4,8)(2,6,5,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,28,30,27)(26,33,31,32)(29,36,37,35)(34,40,38,39);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,18,15,22)(11,19,12,20)(16,23,17,24)(25,29,30,37)(26,34,31,38)(27,35,28,36)(32,39,33,40)

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=hi,e^2=gh,f^2=g,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, 8095688);