#8158966

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

4278 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,e^2=g,f^2=gh,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, 8066162);