#9935980

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

4515 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9843176);