#8933093

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

4569 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=g,e^2=g,f^2=g,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, 8840289);