#7709541

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4232 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)(10,16)(11,13)(17,23)(19,20)(26,29)(27,30)(33,37)(35,36);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,17)(13,18)(15,16)(19,21)(20,24)(22,23)(25,26,31,29)(27,34,30,32)(28,33,38,37)(35,40,36,39);(9,11,15,13)(10,14,16,18)(12,20,22,19)(17,24,23,21)(25,27,31,30)(26,32,29,34)(28,36,38,35)(33,40,37,39);(9,12,15,22)(10,17,16,23)(11,19,13,20)(14,21,18,24)(25,28,31,38)(26,33,29,37)(27,35,30,36)(32,39,34,40);(9,13,15,11)(10,18,16,14)(12,20,22,19)(17,24,23,21)(25,29,31,26)(27,32,30,34)(28,37,38,33)(35,39,36,40);(1,3,4,7)(2,5,6,8)(9,14,15,18)(10,11,16,13)(12,21,22,24)(17,19,23,20)(25,30,31,27)(26,32,29,34)(28,36,38,35)(33,39,37,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,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, 7616737);