#8935950

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

4403 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8843146);