#8127119

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

4512 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)(5,8)(9,10)(11,18)(12,20)(13,15)(14,16)(17,22)(19,23)(21,24)(25,26,29,32)(27,33,34,30)(28,31,36,38)(35,39,40,37);(1,2)(3,5)(4,6)(7,8)(9,11,15,14)(10,16,13,18)(12,21,22,19)(17,24,20,23)(25,26)(27,30)(28,31)(29,32)(33,34)(35,37)(36,38)(39,40);(1,2,4,6)(3,5,7,8)(10,13)(12,22)(16,18)(19,21)(27,34)(28,36)(30,33)(31,38);(1,2)(3,5)(4,6)(7,8)(9,12)(10,17)(11,19)(13,20)(14,21)(15,22)(16,23)(18,24)(25,27)(26,30)(28,35)(29,34)(31,37)(32,33)(36,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,13,15,10)(11,18,14,16)(12,20,22,17)(19,24,21,23)(25,28,29,36)(26,31,32,38)(27,35,34,40)(30,37,33,39);(1,3)(2,5)(4,7)(6,8)(9,14,15,11)(10,18,13,16)(12,21,22,19)(17,24,20,23)(25,26)(27,30)(28,31)(29,32)(33,34)(35,37)(36,38)(39,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,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,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, 8034315);