#7656083

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

4363 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)(11,18)(12,16)(13,15)(14,17)(19,24)(20,22)(21,23)(26,31)(28,35)(32,34)(36,40);(1,2)(3,7)(4,8)(5,6)(9,11,14,13)(10,15,17,18)(12,19,21,20)(16,22,23,24)(25,26)(27,34)(28,33)(29,35)(30,31)(32,38)(36,37)(39,40);(9,11,14,13)(10,15,17,18)(12,19,21,20)(16,22,23,24)(25,27)(26,32)(28,36)(29,37)(30,38)(31,34)(33,39)(35,40);(1,3,5,4)(2,7,6,8)(9,12,14,21)(10,16,17,23)(11,19,13,20)(15,22,18,24)(25,28,30,35)(26,33,31,29)(27,36,38,40)(32,39,34,37);(1,4,5,3)(2,8,6,7)(9,13,14,11)(10,15,17,18)(12,20,21,19)(16,22,23,24)(25,27)(26,34)(28,36)(29,39)(30,38)(31,32)(33,37)(35,40);(9,10)(11,15)(12,16)(13,18)(14,17)(19,22)(20,24)(21,23)(25,29,30,33)(26,35,31,28)(27,37,38,39)(32,40,34,36)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=g,e^2=gi,f^2=i,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, 7563279);