#7927516

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4107 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7834712);