#8728608

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

4464 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8635804);