#7973965

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

4485 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)(11,14)(12,17)(13,16)(15,19)(18,22)(20,24)(21,23)(25,26,31,30)(27,35,36,32)(28,37,38,33)(29,34,39,40);(1,2)(3,7)(4,8)(5,6)(9,10,14,11)(12,15,19,17)(13,18,22,16)(20,24,21,23)(25,27)(26,32)(28,34)(29,37)(30,35)(31,36)(33,39)(38,40);(1,3,5,4)(2,7,6,8)(10,11)(12,19)(16,18)(20,21)(27,36)(29,39)(32,35)(34,40);(1,3,5,4)(2,7,6,8)(9,11,14,10)(12,17,19,15)(13,18,22,16)(20,24,21,23)(25,28,31,38)(26,33,30,37)(27,34,36,40)(29,32,39,35);(9,12)(10,15)(11,17)(13,21)(14,19)(16,24)(18,23)(20,22)(25,29)(26,34)(27,37)(28,32)(30,40)(31,39)(33,36)(35,38);(1,4,5,3)(2,8,6,7)(9,13,14,22)(10,16,11,18)(12,20,19,21)(15,23,17,24)(25,30,31,26)(27,32,36,35)(28,37,38,33)(29,34,39,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=g,f^2=g,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, 7881161);