#7852765

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4322 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)(9,10,15,19)(11,20,23,16)(12,13,24,17)(14,18,22,21)(25,26,31,35)(27,36,39,32)(28,33,40,29)(30,34,38,37);(1,2)(3,7)(4,8)(5,6)(9,11,15,23)(10,16,19,20)(12,21,24,18)(13,22,17,14)(25,27)(26,32)(28,34)(29,38)(30,33)(31,39)(35,36)(37,40);(9,12,15,24)(10,17,19,13)(11,21,23,18)(14,20,22,16)(25,28)(26,33)(27,37)(29,35)(30,36)(31,40)(32,38)(34,39);(1,3,5,4)(2,7,6,8)(9,12)(10,13)(11,21)(14,20)(15,24)(16,22)(17,19)(18,23)(25,29,31,33)(26,28,35,40)(27,38,39,30)(32,34,36,37);(9,13)(10,12)(11,22)(14,23)(15,17)(16,21)(18,20)(19,24)(25,28)(26,33)(27,34)(29,35)(30,32)(31,40)(36,38)(37,39);(1,4,5,3)(2,8,6,7)(9,14,15,22)(10,18,19,21)(11,13,23,17)(12,20,24,16)(25,30,31,38)(26,34,35,37)(27,33,39,29)(28,36,40,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^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, 7759961);