#8348629

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

4663 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)(11,13)(12,20)(14,15)(16,18)(17,21)(19,24)(22,23)(25,26,31,35)(27,32,36,40)(28,29,33,38)(30,34,39,37);(1,2,4,6)(3,5,7,8)(10,15)(11,18)(12,21)(23,24)(25,27)(26,32)(28,30)(29,34)(31,36)(33,39)(35,40)(37,38);(1,3)(2,5)(4,7)(6,8)(10,15)(13,16)(17,20)(23,24)(27,36)(30,39)(32,40)(34,37);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,22)(13,15)(14,18)(17,23)(19,21)(20,24)(25,28)(26,29)(27,30)(31,33)(32,34)(35,38)(36,39)(37,40);(1,3)(2,5)(4,7)(6,8)(9,12)(10,17)(11,19)(13,23)(14,21)(15,20)(16,24)(18,22)(25,29)(26,33)(27,37)(28,35)(30,32)(31,38)(34,36)(39,40);(9,13,14,16)(10,11,15,18)(12,23,21,24)(17,19,20,22)(25,30)(26,34)(27,28)(29,32)(31,39)(33,36)(35,37)(38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,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, 8255825);