#10156564

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

4502 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)(11,18)(12,15)(13,17)(14,16)(19,24)(20,22)(21,23)(25,26)(27,31)(28,33)(29,36)(30,32)(34,40)(35,38)(37,39);(1,2)(3,8)(4,6)(5,7)(9,11,15,14)(10,16,12,18)(13,21,22,19)(17,24,20,23)(25,27,30,35)(26,31,32,38)(28,34,39,36)(29,33,40,37);(1,3,5,4)(2,6,7,8)(10,12)(16,18)(17,20)(23,24)(26,32)(27,35)(33,37)(34,36);(1,2)(3,6)(4,8)(5,7)(9,12,15,10)(11,18,14,16)(13,17,22,20)(19,23,21,24)(25,26)(27,31)(28,37)(29,34)(30,32)(33,39)(35,38)(36,40);(1,4,5,3)(2,6,7,8)(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,28)(26,33)(27,36)(29,31)(30,39)(32,37)(34,35)(38,40);(1,2)(3,6)(4,8)(5,7)(9,14,15,11)(10,18,12,16)(13,19,22,21)(17,23,20,24)(25,29)(26,34)(27,33)(28,38)(30,40)(31,39)(32,36)(35,37)

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=g,e^2=hi,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, 10063760);