#9203718

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

4604 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)(4,5)(6,8)(9,10)(11,15)(12,13)(14,17)(16,18)(19,20)(21,24)(22,23)(25,26,31,35)(27,34,30,29)(28,33,39,32)(36,37,38,40);(2,5)(6,7)(9,11)(10,15)(12,19)(13,20)(14,18)(16,17)(21,22)(23,24)(25,27,31,30)(26,29,35,34)(28,36,39,38)(32,37,33,40);(1,2)(3,6)(4,5)(7,8)(11,18)(15,16)(19,21)(20,24)(25,28,31,39)(26,32,35,33)(27,36,30,38)(29,37,34,40);(1,3,4,8)(2,6,5,7)(9,11)(10,16)(12,21)(13,20)(14,18)(15,17)(19,22)(23,24)(25,28)(26,33)(27,36)(29,40)(30,38)(31,39)(32,35)(34,37);(9,12,14,22)(10,13,17,23)(11,19,18,21)(15,20,16,24)(25,29)(26,27)(28,37)(30,35)(31,34)(32,36)(33,38)(39,40);(9,13)(10,12)(11,20)(14,23)(15,19)(16,21)(17,22)(18,24)(25,30,31,27)(26,34,35,29)(28,38,39,36)(32,40,33,37)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gh,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, 9110914);