#8933770

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	C₂²×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4822 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8840966);