#8806924

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

4500 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=ghi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8714120);