#8648993

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	#37772	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4504 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=ghi,e^2=h,f^2=gi,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, 8556189);