#9717408

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4205 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=g,e^2=hi,f^2=i,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, 9624604);