#10038567

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

4639 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=ghi,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, 9945763);