#9786678

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

4441 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=g,d^2=g,e^2=g,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, 9693874);