#9434198

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

4383 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=hi,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9341394);