#8107221

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

4497 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)(3,6)(4,8)(5,7)(9,10,15,17)(11,16,19,24)(12,14,22,23)(13,21,20,18)(25,26)(27,29)(28,36)(30,31)(32,34)(33,38)(35,39)(37,40);(2,3)(7,8)(9,11)(10,16)(12,13)(14,21)(15,19)(17,24)(18,23)(20,22)(26,31)(27,34)(33,36)(35,40);(2,3)(7,8)(10,17)(11,19)(12,22)(18,21)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,37)(38,40);(1,3,6,2)(4,8,5,7)(9,12)(10,14)(11,20)(13,19)(15,22)(16,18)(17,23)(21,24)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,37)(38,40);(1,4,6,5)(2,7,3,8)(9,13,15,20)(10,18,17,21)(11,12,19,22)(14,16,23,24)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(1,5,6,4)(2,8,3,7)(9,14)(10,12)(11,21)(13,16)(15,23)(17,22)(18,19)(20,24)(25,29,30,32)(26,27,31,34)(28,37,38,39)(33,35,36,40)

Polycyclic presentation:

2^9|a^2=gh,d^2=hi,e^2=h,f^2=g,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, 8014417);