#8943524

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	#2668	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4211 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8850720);