#7657233

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

4370 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)(11,20)(12,17)(13,16)(14,18)(15,19)(21,22)(23,24)(25,26)(27,30)(28,34)(29,37)(31,33)(32,36)(35,38)(39,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,13)(10,16,19,20)(12,21,18,23)(14,22,17,24)(25,27,32,31)(26,33,36,30)(28,37,39,35)(29,34,38,40);(9,11,15,13)(10,16,19,20)(12,21,18,23)(14,22,17,24)(25,28,32,39)(26,34,36,40)(27,35,31,37)(29,30,38,33);(1,3,5,4)(2,7,6,8)(9,12,15,18)(10,17,19,14)(11,21,13,23)(16,24,20,22)(25,29,32,38)(26,35,36,37)(27,34,31,40)(28,30,39,33);(9,13,15,11)(10,16,19,20)(12,23,18,21)(14,22,17,24)(25,30)(26,27)(28,38)(29,39)(31,36)(32,33)(34,35)(37,40);(1,4,5,3)(2,8,6,7)(9,14,15,17)(10,18,19,12)(11,22,13,24)(16,23,20,21)(25,31,32,27)(26,33,36,30)(28,37,39,35)(29,40,38,34)

Polycyclic presentation:

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

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, 7564429);