#8687338

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

4441 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,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, 8594534);