#9712487

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

4368 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9619683);