#9716115

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

4234 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)(3,6,7,4)(9,10,15,14)(11,19,22,16)(12,17,13,18)(20,24,21,23)(25,26)(27,35)(28,29)(30,33)(31,34)(32,38)(36,40)(37,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,20)(13,21)(14,19)(15,22)(17,23)(18,24)(25,27)(26,32)(28,36)(29,37)(30,39)(31,38)(33,40)(34,35);(1,2)(3,4)(5,8)(6,7)(9,12,15,13)(10,17,14,18)(11,20,22,21)(16,23,19,24)(25,28,31,30)(26,33,34,29)(27,36,38,39)(32,40,35,37);(1,4)(2,7)(3,8)(5,6)(9,13,15,12)(10,18,14,17)(11,21,22,20)(16,24,19,23)(25,29,31,33)(26,28,34,30)(27,37,38,40)(32,36,35,39);(1,4)(2,3)(5,6)(7,8)(9,11)(10,19)(12,20)(13,21)(14,16)(15,22)(17,24)(18,23)(25,30,31,28)(26,33,34,29)(27,36,38,39)(32,37,35,40);(1,2)(3,6)(4,7)(5,8)(9,14,15,10)(11,19,22,16)(12,18,13,17)(20,24,21,23)(25,27)(26,32)(28,36)(29,37)(30,39)(31,38)(33,40)(34,35)

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=g,e^2=hi,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, 9623311);