#7895190

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4184 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)(9,10,15,17)(11,18,12,16)(13,22,23,19)(14,20,24,21)(25,26,31,33)(27,35,29,32)(28,34,38,30)(36,37,39,40);(1,2,5,6)(3,7,4,8)(9,11,15,12)(10,16,17,18)(13,20,23,21)(14,19,24,22)(25,27,31,29)(26,32,33,35)(28,36,38,39)(30,37,34,40);(1,3,5,4)(2,7,6,8)(10,17)(11,12)(19,22)(20,21)(26,33)(28,38)(32,35)(36,39);(1,3,5,4)(2,7,6,8)(9,12,15,11)(10,18,17,16)(13,21,23,20)(14,22,24,19)(25,28)(26,34)(27,36)(29,39)(30,33)(31,38)(32,40)(35,37);(9,13,15,23)(10,19,17,22)(11,20,12,21)(14,16,24,18)(25,29,31,27)(26,35,33,32)(28,39,38,36)(30,40,34,37);(1,4,5,3)(2,8,6,7)(9,14)(10,20)(11,19)(12,22)(13,18)(15,24)(16,23)(17,21)(25,30,31,34)(26,28,33,38)(27,37,29,40)(32,36,35,39)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=ghi,e^2=h,f^2=gi,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, 7802386);