#8550926

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	#53064	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4197 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8458122);