#7658079

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

4303 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)(12,19)(17,20)(21,23)(26,27)(31,33)(32,34)(39,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,20)(12,18)(13,17)(14,19)(15,16)(21,22)(23,24)(25,26,30,27)(28,31,36,33)(29,32,38,34)(35,39,37,40);(1,3,4,7)(2,5,6,8)(9,11,15,13)(10,17,16,20)(12,21,19,23)(14,22,18,24)(25,27,30,26)(28,33,36,31)(29,34,38,32)(35,40,37,39);(9,12,15,19)(10,18,16,14)(11,21,13,23)(17,24,20,22);(9,13,15,11)(10,17,16,20)(12,23,19,21)(14,22,18,24)(25,28,30,36)(26,31,27,33)(29,37,38,35)(32,40,34,39);(9,14,15,18)(10,19,16,12)(11,22,13,24)(17,23,20,21)(25,29,30,38)(26,32,27,34)(28,35,36,37)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=i,e^2=hi,f^2=hi,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, 7565275);