#10156579

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

4451 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=g,e^2=ghi,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, 10063775);