#8107313

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

4488 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)(3,6)(4,8)(5,7)(9,10)(11,12)(13,21)(14,15)(16,18)(17,22)(19,24)(20,23)(26,30)(27,33)(32,37)(34,39);(2,3)(7,8)(9,10)(11,18)(12,16)(13,17)(14,15)(19,24)(20,23)(21,22)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(2,3)(7,8)(10,15)(11,16)(17,21)(19,23)(27,33)(31,36)(34,39)(38,40);(1,3,6,2)(4,8,5,7)(9,11,14,16)(10,12,15,18)(13,19,22,23)(17,20,21,24)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,4,6,5)(2,7,3,8)(9,12)(10,16)(11,15)(13,20)(14,18)(17,23)(19,21)(22,24)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40);(1,5,6,4)(2,8,3,7)(9,13,14,22)(10,17,15,21)(11,19,16,23)(12,20,18,24)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40)

Polycyclic presentation:

2^9|d^2=gi,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,d^c=dgi

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, 8014509);