#8159057

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

4286 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)(9,10,15,16)(11,12,13,17)(14,22,23,18)(19,24,21,20)(25,26)(27,34)(28,29)(30,33)(31,37)(32,38)(35,40)(36,39);(1,2,4,6)(3,5,8,7)(10,16)(12,17)(18,22)(20,24)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,13)(10,17,16,12)(14,19,23,21)(18,24,22,20)(25,28,30,32)(26,29,33,38)(27,35,37,39)(31,36,34,40);(1,3,4,8)(2,5,6,7)(9,12,15,17)(10,11,16,13)(14,20,23,24)(18,19,22,21)(25,29,30,38)(26,32,33,28)(27,36,37,40)(31,39,34,35);(9,13,15,11)(10,12,16,17)(14,21,23,19)(18,20,22,24)(25,26)(27,31)(28,29)(30,33)(32,38)(34,37)(35,36)(39,40);(9,14,15,23)(10,18,16,22)(11,19,13,21)(12,20,17,24)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=g,e^2=h,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^d=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, 8066253);