#9779283

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

4330 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=h,f^2=g,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, 9686479);