#10143736

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	#43918	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4216 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=ghi,e^2=gh,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, 10050932);