#9717767

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4247 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=h,e^2=i,f^2=ghi,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, 9624963);