#9127618

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

4642 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9034814);