#7932318

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	#2733	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4183 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)(9,10,15,17)(11,18,14,16)(12,13,22,24)(19,20,21,23)(25,26,32,31)(27,28,34,30)(29,33,39,38)(35,36,40,37);(1,2)(3,5)(4,6)(7,8)(9,11,15,14)(10,16,17,18)(12,21,22,19)(13,20,24,23)(25,27)(26,30)(28,31)(29,35)(32,34)(33,37)(36,38)(39,40);(9,12)(10,13)(11,19)(14,21)(15,22)(16,23)(17,24)(18,20)(25,28,32,30)(26,27,31,34)(29,37,39,36)(33,40,38,35);(9,13)(10,12)(11,20)(14,23)(15,24)(16,21)(17,22)(18,19)(25,29,32,39)(26,33,31,38)(27,35,34,40)(28,36,30,37);(1,3,4,7)(2,5,6,8)(9,12)(10,13)(11,21)(14,19)(15,22)(16,20)(17,24)(18,23)(25,30,32,28)(26,34,31,27)(29,37,39,36)(33,40,38,35);(9,14,15,11)(10,16,17,18)(12,21,22,19)(13,23,24,20)(25,31,32,26)(27,28,34,30)(29,38,39,33)(35,36,40,37)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=h,e^2=gi,f^2=hi,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, 7839514);