#10411734

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

4638 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10318930);