#8119669

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

4490 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)(3,7)(4,6)(5,8)(10,11)(13,21)(15,17)(20,24)(25,26)(27,31)(28,33)(29,37)(30,32)(34,36)(35,39)(38,40);(1,2,4,6)(3,5,8,7)(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(25,27)(26,31)(28,37)(29,33)(30,36)(32,34)(35,38)(39,40);(1,2)(3,5)(4,6)(7,8)(12,19)(15,17)(20,24)(22,23)(26,32)(27,36)(33,40)(35,37);(1,3,4,8)(2,5,6,7)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,28)(26,33)(27,35)(29,34)(30,38)(31,39)(32,40)(36,37);(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,27)(26,34)(28,35)(29,33)(30,36)(31,32)(37,38)(39,40);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,35)(27,33)(28,34)(30,39)(31,38)(32,37)(36,40)

Polycyclic presentation:

2^9|b^2=g,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8026865);