#7961794

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	#43107	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4452 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=gi,e^2=g,f^2=ghi,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, 7868990);