#8509489

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	#38283	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4296 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)(11,17)(12,14)(13,16)(15,19)(18,23)(20,21)(22,24)(25,26,31,33)(27,32,28,36)(29,30,39,40)(34,38,37,35);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,27,31,28)(26,32,33,36)(29,37,39,34)(30,35,40,38);(2,5)(6,8)(13,21)(16,20)(18,24)(22,23)(25,28,31,27)(26,32,33,36)(29,37,39,34)(30,38,40,35);(3,7)(6,8)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,29)(26,30)(27,34)(28,37)(31,39)(32,38)(33,40)(35,36);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,30)(26,29)(27,35)(28,38)(31,40)(32,37)(33,39)(34,36);(1,3)(2,6)(4,7)(5,8)(9,11)(10,17)(12,15)(13,18)(14,19)(16,23)(20,22)(21,24)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=g,e^2=g,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, 8416685);