#7883098

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	#39262	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4422 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)(9,10)(11,16)(12,18)(13,23)(14,20)(15,17)(19,22)(21,24)(25,26)(27,36)(28,29)(30,34)(31,33)(32,35)(37,38)(39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,22)(13,20)(14,23)(15,21)(17,24)(18,19)(25,27,32,31)(26,33,35,36)(28,39,30,37)(29,38,34,40);(10,17)(11,21)(12,20)(19,23)(25,28,32,30)(26,29,35,34)(27,37,31,39)(33,40,36,38);(1,2,5,4)(3,6,8,7)(9,12,15,20)(10,18,17,14)(11,22,21,13)(16,19,24,23)(25,29,32,34)(26,28,35,30)(27,38,31,40)(33,39,36,37);(1,4,5,2)(3,7,8,6)(9,13,15,22)(10,19,17,23)(11,20,21,12)(14,16,18,24)(25,30,32,28)(26,34,35,29)(27,37,31,39)(33,40,36,38);(9,14,15,18)(10,20,17,12)(11,23,21,19)(13,24,22,16)(25,31,32,27)(26,33,35,36)(28,39,30,37)(29,40,34,38)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=gh,e^2=gh,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7790294);