#7736553

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

4484 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:

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

Polycyclic presentation:

2^9|a^2=ghi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7643749);