#10155820

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

4472 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,e^2=i,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, 10063016);