#8587744

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	#43491	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4578 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^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, 8494940);