#9815331

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

4505 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)(3,8)(4,7)(5,6)(9,10,15,20)(11,16,22,13)(12,18,24,17)(14,23,21,19)(26,31)(27,34)(29,37)(35,39);(2,6)(7,8)(9,11,15,22)(10,16,20,13)(12,21,24,14)(17,23,18,19)(25,26,30,31)(27,32,34,38)(28,29,33,37)(35,36,39,40);(9,12,15,24)(10,17,20,18)(11,14,22,21)(13,23,16,19)(25,27,30,34)(26,32,31,38)(28,35,33,39)(29,36,37,40);(1,3,5,4)(2,7,6,8)(9,12)(10,18)(11,21)(13,23)(14,22)(15,24)(16,19)(17,20)(25,27)(26,32)(28,35)(29,36)(30,34)(31,38)(33,39)(37,40);(1,2)(3,7)(4,8)(5,6)(9,13)(10,11)(12,23)(14,17)(15,16)(18,21)(19,24)(20,22)(25,28)(26,29)(27,35)(30,33)(31,37)(32,36)(34,39)(38,40);(1,4,5,3)(2,7,6,8)(9,14,15,21)(10,19,20,23)(11,12,22,24)(13,18,16,17)(25,29)(26,33)(27,36)(28,31)(30,37)(32,39)(34,40)(35,38)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=ghi,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, 9722527);