#9726962

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
prime radical	#50175	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4269 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=gi,f^2=h,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, 9634158);