#8366604

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

4364 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)(3,8,5,7)(9,10,14,15)(11,17,18,13)(12,20,21,16)(19,23,22,24);(1,3,6,5)(2,7,4,8)(10,15)(11,18)(12,21)(23,24)(25,26,30,31)(27,32,36,38)(28,35,39,33)(29,34,40,37);(1,2,6,4)(3,7,5,8)(10,15)(13,17)(16,20)(23,24)(26,31)(29,40)(32,38)(33,35);(1,3,6,5)(2,7,4,8)(9,11)(10,13)(12,22)(14,18)(15,17)(16,24)(19,21)(20,23)(25,27)(26,32)(28,37)(29,35)(30,36)(31,38)(33,40)(34,39);(1,4,6,2)(3,7,5,8)(9,12)(10,16)(11,19)(13,23)(14,21)(15,20)(17,24)(18,22)(25,28,30,39)(26,33,31,35)(27,34,36,37)(29,38,40,32);(1,5,6,3)(2,7,4,8)(9,13)(10,11)(12,23)(14,17)(15,18)(16,19)(20,22)(21,24)(25,29)(26,34)(27,35)(28,38)(30,40)(31,37)(32,39)(33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8273800);