#7652770

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

4391 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)(9,10,14,13)(11,17,19,15)(12,16,22,21)(18,24,20,23);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,23)(20,22)(21,24)(25,26,30,31)(27,29,33,38)(28,32,37,39)(34,36,35,40);(1,3,4,7)(2,6,5,8)(10,13)(15,17)(16,21)(23,24)(26,31)(29,38)(32,39)(36,40);(9,11)(10,15)(12,20)(13,17)(14,19)(16,24)(18,22)(21,23)(25,27,30,33)(26,29,31,38)(28,35,37,34)(32,40,39,36);(9,12,14,22)(10,16,13,21)(11,18,19,20)(15,23,17,24)(25,28,30,37)(26,32,31,39)(27,34,33,35)(29,36,38,40);(9,13,14,10)(11,17,19,15)(12,21,22,16)(18,24,20,23)(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=i,c^2=gh,d^2=i,e^2=hi,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, 7559966);