#9203716

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

4583 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)(11,16,19,23)(12,17,22,21)(13,20,24,18)(25,26)(27,36)(28,32)(29,33)(30,34)(31,35)(37,38)(39,40);(2,5)(3,8)(10,15)(12,22)(16,23)(18,20)(25,27,31,34)(26,30,35,36)(28,37,33,40)(29,38,32,39);(1,3)(2,6)(4,5)(7,8)(10,15)(11,19)(17,21)(18,20)(25,28,31,33)(26,32,35,29)(27,37,34,40)(30,39,36,38);(1,4,7,6)(2,3,5,8)(9,11,14,19)(10,16,15,23)(12,18,22,20)(13,21,24,17)(25,28)(26,32)(27,37)(29,35)(30,39)(31,33)(34,40)(36,38);(1,5,7,2)(3,4,8,6)(9,12)(10,17)(11,20)(13,23)(14,22)(15,21)(16,24)(18,19)(25,29,31,32)(26,33,35,28)(27,38,34,39)(30,40,36,37);(1,6,7,4)(2,8,5,3)(9,13,14,24)(10,18,15,20)(11,21,19,17)(12,16,22,23)(25,30)(26,34)(27,35)(28,39)(29,37)(31,36)(32,40)(33,38)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=h,e^2=ghi,f^2=h,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, 9110912);