#7652783

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

4538 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=i,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

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, 7559979);