#10042514

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

4412 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9949710);