#9236482

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

4564 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:

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

Polycyclic presentation:

2^9|d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9143678);