#8399114

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
prime radical	#40116	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4410 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8306310);