#8108389

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

4646 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015585);