#10166300

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

4300 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=hi,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, 10073496);