#10068960

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

4426 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9976156);