#8599522

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
prime radical	#49556	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4468 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)(11,22)(12,19)(13,18)(14,23)(15,17)(16,21)(20,24)(25,26,31,35)(27,29,36,32)(28,33,37,38)(30,34,39,40);(2,5)(6,8)(9,11,16,15)(10,17,21,22)(12,23,13,20)(14,19,24,18)(27,36)(28,37)(29,32)(33,38);(1,2,4,5)(3,6,7,8)(9,12,16,13)(10,18,21,19)(11,23,15,20)(14,22,24,17)(25,27,31,36)(26,32,35,29)(28,30,37,39)(33,40,38,34);(3,7)(6,8)(9,13,16,12)(10,19,21,18)(11,23,15,20)(14,22,24,17)(25,28)(26,33)(27,30)(29,34)(31,37)(32,40)(35,38)(36,39);(1,2)(3,6)(4,5)(7,8)(9,14,16,24)(10,20,21,23)(11,18,15,19)(12,17,13,22)(25,29,31,32)(26,27,35,36)(28,34,37,40)(30,38,39,33);(1,3)(2,6)(4,7)(5,8)(9,15,16,11)(10,17,21,22)(12,20,13,23)(14,19,24,18)(25,30)(26,34)(27,37)(28,36)(29,38)(31,39)(32,33)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8506718);