#8932609

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

4433 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=g,e^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,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, 8839805);