#7969651

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4397 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)(9,10)(11,14)(12,20)(13,17)(15,19)(16,22)(18,24)(21,23)(25,26,31,35)(27,36,38,32)(28,33,40,30)(29,39,37,34);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,23)(20,24)(21,22)(25,27,31,38)(26,32,35,36)(28,39,40,34)(29,30,37,33);(9,11,13,19)(10,15,17,14)(12,21,22,18)(16,23,20,24)(25,28)(26,33)(27,34)(29,36)(30,35)(31,40)(32,37)(38,39);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,24)(20,23)(21,22)(25,29,31,37)(26,34,35,39)(27,30,38,33)(28,36,40,32);(9,12)(10,16)(11,18)(13,22)(14,23)(15,24)(17,20)(19,21)(25,30)(26,28)(27,37)(29,38)(31,33)(32,39)(34,36)(35,40);(9,10)(11,15)(12,20)(13,17)(14,19)(16,22)(18,23)(21,24)(25,28)(26,33)(27,34)(29,36)(30,35)(31,40)(32,37)(38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7876847);