#9625469

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	#55514	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4097 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9532665);