#8362453

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

4376 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=h,e^2=gi,f^2=h,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, 8269649);