#9203688

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

4519 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,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, 9110884);