#8599555

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	#38564	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4549 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=g,e^2=gh,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, 8506751);