#10156887

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) = 128

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=gh,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, 10064083);