#7862642

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	#35571	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4518 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=h,e^2=gh,f^2=g,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, 7769838);