#7653086

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4344 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=g,e^2=i,f^2=i,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, 7560282);