#8926181

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

4532 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833377);