#7658669

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

4457 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565865);