#10158322

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

4494 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)(11,16)(12,13)(14,22)(15,17)(18,24)(19,21)(20,23)(25,26,29,31)(27,33,35,30)(28,32,38,37)(34,40,36,39);(1,2)(3,8)(4,6)(5,7)(9,11,15,12)(10,16,17,13)(14,19,24,20)(18,23,22,21)(25,27)(26,30)(28,36)(29,35)(31,33)(32,40)(34,38)(37,39);(1,3,5,4)(2,6,7,8)(10,17)(11,12)(18,22)(19,20)(26,31)(30,33)(32,37)(39,40);(1,2)(3,6)(4,8)(5,7)(9,12,15,11)(10,13,17,16)(14,20,24,19)(18,21,22,23)(25,26)(27,30)(28,37)(29,31)(32,38)(33,35)(34,40)(36,39);(1,4,5,3)(2,6,7,8)(9,13)(10,12)(11,17)(14,23)(15,16)(18,19)(20,22)(21,24)(25,28)(26,32)(27,34)(29,38)(30,39)(31,37)(33,40)(35,36);(1,2)(3,6)(4,8)(5,7)(9,14,15,24)(10,18,17,22)(11,19,12,20)(13,21,16,23)(25,27)(26,30)(28,36)(29,35)(31,33)(32,40)(34,38)(37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=gh,e^2=ghi,f^2=gh,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, 10065518);