#7657545

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

4201 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)(9,10,15,19)(11,17,12,16)(13,14,18,24)(20,23,22,21)(25,26)(27,30)(28,36)(29,33)(31,34)(32,39)(35,37)(38,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,12)(10,16,19,17)(13,20,18,22)(14,21,24,23)(25,27,31,33)(26,30,34,29)(28,35,39,40)(32,38,36,37);(1,3,4,7)(2,5,6,8)(9,12,15,11)(10,17,19,16)(13,22,18,20)(14,23,24,21)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,38)(39,40);(25,28,31,39)(26,32,34,36)(27,35,33,40)(29,37,30,38);(9,13)(10,14)(11,20)(12,22)(15,18)(16,21)(17,23)(19,24)(25,29,31,30)(26,33,34,27)(28,37,39,38)(32,40,36,35);(9,14)(10,18)(11,21)(12,23)(13,19)(15,24)(16,22)(17,20)(25,30,31,29)(26,27,34,33)(28,38,39,37)(32,35,36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=i,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,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, 7564741);