#9061598

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	#35729	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4503 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)(11,16)(12,18)(13,19)(14,23)(15,17)(20,22)(21,24)(25,26,31,35)(27,36,39,32)(28,30,38,33)(29,34,37,40);(1,2,5,6)(3,7,8,4)(9,11,15,14)(10,16,17,23)(12,20,21,19)(13,18,22,24)(25,27)(26,32)(28,37)(29,38)(30,34)(31,39)(33,40)(35,36);(1,2)(3,7)(4,8)(5,6)(10,17)(11,14)(12,21)(13,22)(25,28,31,38)(26,33,35,30)(27,37,39,29)(32,40,36,34);(2,6)(3,8)(9,12,15,21)(10,18,17,24)(11,19,14,20)(13,23,22,16)(25,28)(26,33)(27,29)(30,35)(31,38)(32,34)(36,40)(37,39);(1,3)(2,4)(5,8)(6,7)(9,13)(10,19)(11,18)(12,16)(14,24)(15,22)(17,20)(21,23)(25,29)(26,34)(27,38)(28,39)(30,32)(31,37)(33,36)(35,40);(1,4)(2,3)(5,7)(6,8)(9,14,15,11)(10,16,17,23)(12,20,21,19)(13,24,22,18)(25,30,31,33)(26,28,35,38)(27,34,39,40)(29,32,37,36)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8968794);