#8933214

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

4413 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=g,e^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, 8840410);