#7644318

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	#42928	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4220 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:

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

Polycyclic presentation:

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

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, 7551514);