#9167447

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

4531 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=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, 9074643);