#10158252

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
prime radical	#38521	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4215 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)(11,19)(12,22)(13,18)(14,16)(15,17)(20,24)(21,23)(25,26,31,32)(27,29,34,39)(28,33,37,40)(30,38,36,35);(1,2)(3,8)(4,6)(5,7)(9,11,15,14)(10,16,17,19)(12,20,18,23)(13,24,22,21)(25,27)(26,29)(28,36)(30,37)(31,34)(32,39)(33,35)(38,40);(1,3,5,4)(2,6,7,8)(10,17)(12,18)(16,19)(20,23)(26,32)(27,34)(28,37)(35,38);(1,2)(3,6)(4,8)(5,7)(9,12)(10,13)(11,20)(14,23)(15,18)(16,24)(17,22)(19,21)(25,28,31,37)(26,33,32,40)(27,36,34,30)(29,35,39,38);(1,4,5,3)(2,6,7,8)(9,13)(10,18)(11,21)(12,17)(14,24)(15,22)(16,20)(19,23)(25,29)(26,34)(27,32)(28,38)(30,40)(31,39)(33,36)(35,37);(1,2)(3,6)(4,8)(5,7)(9,14,15,11)(10,19,17,16)(12,23,18,20)(13,21,22,24)(25,30,31,36)(26,35,32,38)(27,37,34,28)(29,33,39,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=gh,e^2=ghi,f^2=h,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, 10065448);