#7989661

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4263 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=gi,e^2=gi,f^2=ghi,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, 7896857);