#9789183

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

4520 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9696379);