#10171868

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

4423 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)(3,5)(4,6)(7,8)(9,10,15,14)(11,17,12,16)(13,18,21,22)(19,24,20,23)(25,26,30,33)(27,32,36,31)(28,29,39,37)(34,40,38,35);(1,3,7,6)(2,4,8,5)(9,11,15,12)(10,16,14,17)(13,19,21,20)(18,23,22,24)(25,26,30,33)(27,32,36,31)(28,37,39,29)(34,35,38,40);(1,4,7,5)(2,3,8,6)(25,27,30,36)(26,31,33,32)(28,34,39,38)(29,35,37,40);(2,8)(4,5)(9,12,15,11)(10,17,14,16)(13,19,21,20)(18,23,22,24)(25,27)(26,32)(28,38)(29,35)(30,36)(31,33)(34,39)(37,40);(1,5,7,4)(2,3,8,6)(9,13,15,21)(10,18,14,22)(11,19,12,20)(16,23,17,24)(25,28,30,39)(26,29,33,37)(27,34,36,38)(31,35,32,40);(1,6,7,3)(2,5,8,4)(9,14,15,10)(11,17,12,16)(13,18,21,22)(19,23,20,24)(25,29)(26,28)(27,35)(30,37)(31,34)(32,38)(33,39)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10079064);