#9815091

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

4323 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)(3,4)(5,6)(7,8)(9,10)(11,20)(12,14)(13,16)(15,17)(18,19)(21,24)(22,23)(25,26,31,33)(27,32,36,29)(28,34,38,40)(30,39,37,35);(2,6)(4,7)(9,11,15,13)(10,16,17,20)(12,21,19,23)(14,22,18,24)(25,27,31,36)(26,32,33,29)(28,37,38,30)(34,35,40,39);(10,17)(12,19)(16,20)(21,23)(26,33)(27,36)(28,38)(35,39);(1,3)(2,7)(4,6)(5,8)(9,12,15,19)(10,18,17,14)(11,21,13,23)(16,24,20,22)(25,28)(26,34)(27,37)(29,39)(30,36)(31,38)(32,35)(33,40);(1,4,5,7)(2,3,6,8)(9,13,15,11)(10,16,17,20)(12,23,19,21)(14,22,18,24)(25,29)(26,27)(28,39)(30,40)(31,32)(33,36)(34,37)(35,38);(1,3)(2,4)(5,8)(6,7)(9,14,15,18)(10,19,17,12)(11,22,13,24)(16,23,20,21)(25,30,31,37)(26,35,33,39)(27,28,36,38)(29,34,32,40)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=hi,e^2=g,f^2=h,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, 9722287);