#8399305

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

4530 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

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, 8306501);