#10045520

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

4509 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)(3,7)(4,6)(5,8)(9,10,14,15)(11,16,19,22)(12,17,18,13)(20,23,24,21)(25,26,30,32)(27,31,34,40)(28,37,33,29)(35,36,38,39);(1,3,5,4)(2,6,8,7)(10,15)(11,19)(12,18)(21,23)(25,27,30,34)(26,31,32,40)(28,38,33,35)(29,36,37,39);(1,3,5,4)(2,7,8,6)(10,15)(11,19)(12,18)(21,23)(26,32)(27,34)(28,33)(36,39);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,20)(13,21)(14,19)(15,22)(17,23)(18,24)(25,28,30,33)(26,29,32,37)(27,35,34,38)(31,39,40,36);(1,2)(3,6)(4,7)(5,8)(9,12)(10,17)(11,20)(13,15)(14,18)(16,23)(19,24)(21,22)(25,27)(26,31)(28,35)(29,39)(30,34)(32,40)(33,38)(36,37);(1,4,5,3)(2,6,8,7)(9,13,14,17)(10,18,15,12)(11,21,19,23)(16,24,22,20)(25,29)(26,33)(27,36)(28,32)(30,37)(31,35)(34,39)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=ghi,d^2=i,f^2=g,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, 9952716);