#8760044

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4140 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8667240);