#7989667

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	#51061	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4418 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)(11,16,12,20)(13,23,24,14)(18,19,21,22)(25,26,31,35)(27,32,30,34)(28,29,39,33)(36,37,38,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,12)(10,16,17,20)(13,18,24,21)(14,22,23,19)(25,27,31,30)(26,32,35,34)(28,36,39,38)(29,37,33,40);(10,17)(11,12)(13,24)(19,22)(25,28,31,39)(26,33,35,29)(27,36,30,38)(32,40,34,37);(1,3,5,4)(2,7,6,8)(9,12,15,11)(10,16,17,20)(13,21,24,18)(14,22,23,19)(25,28)(26,29)(27,36)(30,38)(31,39)(32,37)(33,35)(34,40);(1,3,5,4)(2,7,6,8)(9,13,15,24)(10,14,17,23)(11,18,12,21)(16,22,20,19)(25,29)(26,28)(27,37)(30,40)(31,33)(32,36)(34,38)(35,39);(1,4,5,3)(2,8,6,7)(9,14)(10,13)(11,19)(12,22)(15,23)(16,21)(17,24)(18,20)(25,30,31,27)(26,34,35,32)(28,38,39,36)(29,40,33,37)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=gi,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7896863);