#7647067

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

4385 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7554263);