#7655550

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

4477 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:

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

Polycyclic presentation:

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

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, 7562746);