#7660655

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
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4257 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)(25,26,31,35)(27,28,37,32)(29,33,36,38)(30,34,39,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22);(10,14)(15,21)(16,22)(23,24)(25,27,31,37)(26,32,35,28)(29,30,36,39)(33,40,38,34);(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,28,31,32)(26,27,35,37)(29,34,36,40)(30,38,39,33);(1,3,4,7)(2,5,6,8)(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(25,29)(26,33)(27,30)(28,34)(31,36)(32,40)(35,38)(37,39);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(25,30)(26,34)(27,36)(28,38)(29,37)(31,39)(32,33)(35,40)

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=h,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,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, 7567851);